
Class Q B44- 
Book. ■ J3 



CDFHUGHT DEPOSIT. 



ASTRONOMY 






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THE MACMILLAN COMPANY 

NEW YORK t BOSTON • CHICAGO • DALLAS 
ATLANTA • SAN FRANCISCO 

MACMILLAN & CO., Limited 

LONDON • BOMBAY • CALCUTTA 
MELBOURNE 

THE MACMILLAN CO. OF CANADA, Ltd. 

TORONTO 




PLATE 1. 



Photo by Barnard, Oct. 19, 1911. Exposure, 46 mi/i. (see p. 14). 

Comet c 1911, discovered by Brooks. 



ASTKONOMY 



A POPULAR HANDBOOK 



BY 

HAROLD JACOBT 

RUTHERFURD PROFESSOR OF ASTRONOMY 
IN COLUMBIA UNIVERSITY 



WITH THIRTY-TWO PLATES AND MANY FIGURES 
IN THE TEXT 



ETefo got* 

THE MACMILLAN COMPANY 
1913 

All rights reserved 



Q B A-A- 



Copyright, 1913, 
By THE MACMILLAN COMPANY. 



Set up and electrotyped. Published September, 1913, 



• 



'.< 



Nortoooti $«»» 

J. S. Cushing Co. — Berwick & Smith Co. 

Norwood, Mass., U.S.A. 






" To know 
That which before us lies in daily life 
Is the prime wisdom." 

— Paradise Lost, VIII, 192. 



PREFACE 

The present volume has been prepared with a double 
purpose, and upon a plan somewhat unusual. First, an 
effort has been made to meet the wishes of the ordinary 
reader who may desire to inform himself as to the present 
state of astronomic science, or to secure a simple explanation 
of the many phenomena constantly exhibiting themselves 
in the universe about him ; and the further purpose has been 
to produce a satisfactory textbook for use in high schools 
and colleges. 

Thus, for the general reader, it has been thought 
necessary to eliminate all formal mathematics ; for the 
student, on the other hand, the occasional use of elementary 
algebra and geometry are essential. To satisfy these two 
apparently contradictory conditions, the book has been 
written in two parts ; the first free from mathematics, the 
second a series of extended elementary mathematical notes 
and explanations to which appropriate references are made 
in the first part of the book. Thus the general reader may 
confine his attention to the non-mathematical part ; the 
student should master the whole volume. 

Attention is directed especially to Chapter I, in which is 
presented a brief summary of the entire science. It is 
hoped that this will serve to strengthen in most readers a 
desire for further and more detailed information. To the 
student this chapter should furnish as much knowledge 
as he must have in his possession before beginning a direct 

vii 



PREFACE 

study of the sky with a telescope. In the author's extended 
experience as a college teacher of elementary astronomy, he 
has found it most desirable to give life to the subject by 
requiring frequent evening visits of students at the obser- 
vatory. These should begin almost immediately upon 
commencing the study of the science ; and the first chapter 
is therefore intended to give the students something to work 
upon, even in their earliest observatory visits. At Columbia 
and Barnard colleges, these visits are required on frequent 
dates, regularly assigned throughout the year, without 
regard to the state of the weather. When clear, the tele- 
scope is used ; when this is impossible, oral and informal 
discussion takes place upon the work done in the classroom. 
Attention is also given to the daylight study of solar 
shadows, all students being required to construct a prac- 
tical sundial, as explained in Chapter V. 

The author has, of course, drawn freely upon many othei 
books, especially in the preparation of numerous diagrams, 
and in arranging the various parts of the subject in order. 
But most of the diagrams are new, and all have been sim- 
plified as much as possible. In a few cases, illustrations 
were copied from very old astronomic textbooks: references 
are then always given, in the hope that some readers, at 
least, will be led to examine these fine venerable classics of 
the science. 

Almost all the inserted plates are photographic repro- 
ductions of actual photographs. For these the author is 
under deep obligations to Professor E. E. Barnard, of the 
Yerkes Observatory, and to the astronomers of the Lick 
Observatory. g J. 

Columbia University, 
May, 1913. 

Vlll 



TABLE OF CONTENTS 

CHAPTEE PAGE 

I. The Universe 1 

Introductory. General view of the science. Its practical 
use: navigation, coast and boundary surveying, timekeeping 
for mankind. Value as a culture study. 

II. The Heavens 22 

What we can see by examining the sky without a tele- 
scope. The celestial sphere with its points, lines, and cir- 
cles. Diurnal phenomena; day and night; rising and setting 
of the stars. Aspect of the heavens from New York, from 
the equator, and from the polar regions. 

III. HOW TO KNOW THE STARS 45 

The planets and the principal fixed stars and constella- 
tions. Maps, globes, and planispheres. 

IV. Time 65 

Star-time or sidereal time. Solar time and standard time. 
Differences of time between different places on the earth. 
The international date line. 
V. The Sundial 78 

How to make one, and how to use it. 

VI. Mother Earth 86 

Notions of the ancients. Proof of curvature and rotation. 
The Foucault experiment. Measurement of the earth's size 
and shape ; by the ancients, and by the moderns. Geodesy. 
The earth's mass: weighing the earth. Experiments of 
Maskelyne and Cavendish. The terrestrial interior. Varia- 
tion of latitudes. The atmosphere : twilight ; refraction. 

VLI. The Earth in Relation to the Sun 116 

Orbit around the sun : how it might be determined. The 
seasons. Astronomic explanation of the geologic ice age. 
The length of the year determined by the ancients. Trop- 
ical and sidereal years. Precession of the equinoxes. Nu- 
tation. Age of the great pyramid. Equation of time. 
Aberration of light ; its discovery by Bradley, 
ix 



TABLE OF CONTENTS 

CHAPTER PAGB 

VIII. The Calendar 138 

History of the calendar. How to find the day of the 
week for any date, past or future. Perpetual calendars : 
how to make and use them. How to find the date of 
Easter Sunday in any year. 

IX. Navigation . 151 

How ships find their way across the ocean. Method 
used before the days of chronometers. 

X. Moonshine 160 

Source of the moon's light. The lunar months, sidereal 
and synodic. Phases of the moon. Phases of the earth. 
Air and water absent on the moon. Occultations. Meas- 
urement of the distance of the moon from the earth. Axial 
rotation. Librations. Determination of lunar diameter, 
volume and weight. Lunar day. Harvest moon. Sun 
and moon in the almanac. Moon's true orbit. Measure- 
ment of the height of lunar mountains. 

XL The Planets 183 

Kepler and Newton. Central forces. Ptolemaic theory 
and Copernican theory. Planetary periods. Bode's law. 
Modern orbit work. Elements of orbits. Measuring and 
weighing planets. Satellites. Mechanical stability and 
perturbations in the solar system. Conjunctions and oppo- 
sitions : visibility of planets. 

XII. The Planets One by One 217 

Each planet's characteristics considered separately. Hab- 
itability of Mars. 

XIII. The Tides . 251 

Explanation of tidal phenomena due to lunar attraction 
modified by solar attraction. Effect of the tides on the 
moon itself. 

XTV. The Solar Parallax 260 

Distance from the earth to the sun. Modern investiga- 
tions : minor planet method; Eros; transit of Venus; 
Halley's method ; indirect methods. 
x 



TABLE OF CONTENTS 

CHAPTER PAGE 

XV. Astronomic Instruments 272 

The telescope : magnifying power ; cross-threads and mi- 
crometers. The meridian circle and chronograph. The 
equatorial. Photographic telescopes. The spectroscope. 

XVI. Sunshine 286 

Constitution of the sun. Sunspots. Measuring and 
weighing the sun. Theories as to durability of the sun. 
Photosphere, chromosphere and corona. Axial rotation of 
the sun. 



XVII. 



XVIII. 



XIX. 



XX. 



XXI. 



Eclipses 297 

Explanation of their cause. Eclipse limits. Umbra 
and penumbra. Annular eclipses. Prediction of eclipses 
by means of the Saros. Transits of Mercury and Venus. 

Comets 307 

The coma, nucleus and tail. Size and mass. Danger of 
collision with the earth. Light pressure theory. Comet 
hunting. Naming comets. Their orbits. Families and 
relationships of comets. 



Meteors and Aerolites 

Shooting stars. Showers. Radiant. Cause of light. 
Fragments of comets. Height above the earth's surface. 
Aerolites : their chemical composition. 

Starshine 

Magnitude and brilliancy of the stars. Variable and 
temporary stars. Stellar eclipses. Star distances : meas- 
urement of parallax ; the light-year. Motions of the fixed 
stars : proper motion and radial velocity. Stellar chemis- 
try. The sun's own motion in space : the apex. Shall 
we reach Vega? Statistical studies of the universe. Stel- 
lar distribution. Kinetic theory of stars. Binary stars. 
Clusters and nebulae. The galaxy. 

The Universe once More 

Origin of the universe. Laplace's nebular hypothesis. 
Chamberlin's planetesimal hypothesis. • 



315 



322 



Appendix 



356 



363 



Elementary mathematical explanations. 



xi 



LIST OF PLATES 



1. Comet c 1911, discovered by Brooks 



9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 



Spiral Nebula ..... 

The Moon in the First Quarter Phase 

The Samrat Yantra, " Prince of Dials, 

Precessional Motion of the Pole 

Full Moon and Crescent Moon 

Lunar Enlargement .... 

Mars and the Crescent Venus . 

Discovery of Planetoids 

Saturn 

Saturn 

The Lick Observatory, Mt. Hamilton, Cal 

The Lick Telescope . 

The Crossley Reflector 

Lick Spectroscope . 

Various Spectra 

The Sun . 

Great Sunspot . 

The Prominences 

Total Solar Eclipse, with Corona 

The Morehouse Comet, Nov. 18, 

Halley's Comet 

Meteor Trail .... 

The Constellation Serpentarius 

Nova Persei .... 

Spectra 

A Star Cluster in Hercules and 
The Pleiades .... 
The North America Nebula 

Nebulse 

The Trifid Nebula . 
Title-page of Newton's Principia 



at Jaipur 




Frontispiece / 

TO FACE PAGE 

4 
. 17 

. 84/ 

. 133^ 

. 162 l/ 

. 182/ 

. 225 

. 234 

. 242 

. 244/ 

. 272 

. 279 . 

. 281 

. 283. 

. 284 

. 288 

. 290 

. 293 

. 295 

. 307 

. 311 

. 318. 

. 323 

. 328 

. 337 

. 349 

. 351 

. 354 

. 359 J 

. 360 

. 399 



t 



ASTRONOMY 



CHAPTER I 

THE UNIVERSE 

If a company of men and women should chance to be 
gathered together on some clear, quiet evening under the 
dome of the sky, and if there should happen to come into 
that company one known to possess an acquaintance with 
the facts and the theories of astronomic science, — if these 
things should occur, inevitably there would descend upon 
that astronomer a shower of questions. These he would 
answer in simple language, after a kindly fashion for many 
centuries the habit of his guild ; and, as he passed on, he 
would once more marvel, as he had done many times before, 
at the changelessness of man's desires. For these ques- 
tions of the multitude, welcome ever to the star-man, to-day 
still resemble those that were laid of old as problems before 
his predecessors at the side of the pyramids. 

Why, for instance, does the moon appear at times as a full 
round disk, at others as a tiny crescent ? Why do certain 
bright stars called planets seem to wander about among 
the multitude of their fellows? Why is summer hot and 
winter cold ? How do navigators find their way across the 
trackless ocean by observing the heavenly bodies ? 

Let us begin by attempting to set forth as best we may 
the answer to some of these many eternal questions from the 
skies. For the astronomer is not always present ; and even 
B l 



ASTRONOMY 

if he were, it is often better to gather our information in 
silence, by means of the process called reading. The very 
name Astronomy tells us what our science is. Derived 
from two Greek words, Astronomy means "the law of the 
stars." Where does the law of the stars hold sway? 
Throughout all space. What is space ? Space is the place 
where astronomy has its being. When does astronomy 
enforce its laws? Throughout all time. What is time? 
Time is the period during which astronomy has its being. 
Astronomy needs no logical definitions of space and time. 
They belong to it ; they are part of it. 

Somewhere, then, in the endless void of space our universe 
is suspended ; the visible universe. Is that visible universe 
but one of many? Are there invisible universes without 
number scattered through the vastness of space like conti- 
nents in an endless ocean? The human mind loses itself 
in speculations such as this : nor do such speculations here 
concern us ; for astronomers consider only the ascertainable 
phenomena of the universe that unfolds itself to our senses. 

There is in existence a vast quantity of matter and a vast 
quantity of active energy, or force. It is not necessary 
at this point to define these terms ; but we should remember 
that according to accepted theory the total of matter and 
the total of energy in the universe do not change. Matter 
is never destroyed ; and the accepted law of the conservation 
of energy tells us that the quantity of energy in the universe 
is likewise constant and unvarying in amount. None ever 
disappears out of existence. But both matter and energy 
may and do undergo changes in form and appearance. 
Thus, water may appear as steam or as ice ; and the energy 
of a moving body may be transformed into heat, light, or 
electricity. 



THE UNIVERSE 

When we examine this visible universe of ours at night 
with the unaided eye, we see several different kinds of ob- 
jects : nebulae, or small luminous clouds ; star clusters, like 
the famous group called the Pleiades ; individual stars ; the 
moon ; and, occasionally, comets or meteors. In the day 
we see the sun ; sometimes the moon ; and very rarely 
indeed a particularly bright star or comet. We shall give 
here a brief outline of existing knowledge concerning these 
various celestial objects, leaving a detailed description of 
their peculiarities to later chapters. They are all com- 
posed of matter ; all, if in motion, move in accordance with 
the laws of mechanical science which govern the operation 
of energy ; and all, if they change, undergo only changes 
such as accord with the laws of physics and chemistry. 

First, then, the nebulae. We shall begin with these 
because they probably represent the form in which matter 
shows itself to us in its most primitive stage of development. 
Only one or two can be seen with the unaided eye; and 
these only on very clear nights when the moon is invis- 
ible. In the telescope they appear as patches of luminous 
cloud, often more or less irregular in form. They were 
once thought to be simply conglomerations of small stars, 
so close to each other that the optical powers of existing 
telescopes were unable to separate them into constituent 
units. This view gained in probability for a long time, 
because, as the power of telescopes increased with the in- 
crease of skill among opticians, astronomers were con- 
stantly resolving new nebulae, as they used to call it ; sepa- 
rating them into simple close clusters of faint stars. 

But the invention of an instrument called the spec- 
troscope, in the middle of the nineteenth century, put us 
in possession of a means, previously non-existent, for dis- 

3 



ASTRONOMY 

tinguishing with certainty between the light of incandescent 
gases and that derived from incandescent or luminous matter 
in the liquid or solid stage. With the spectroscope astrono- 
mers have been able to ascertain that there are many nebulae 
in a truly gaseous condition; that probably most of these 
objects are gaseous bodies ; that they could not be resolved 
into stars, even if terrestrial man possessed to-day tele- 
scopes more powerful than he is likely ever to have at his 
command. 

According to many modern theorists, we may take the 
nebulae to be matter not yet fashioned into stars. This 
means, of course, that certain forces are at work in the 
nebulae ; forces of irresistible power, slow in action, as all 
cosmic changes must be slow when measured by the life of 
human generations ; but sure in action, too, with that in- 
finite sureness which belongs in celestial spaces. These 
forces doubtless produce motions of vast import within the 
body of the nebula ; heat is doubtless engendered ; conden- 
sations occur at certain points ; nuclei are formed ; prob- 
ably, finally, one or more stars take the place of these nuclei ; 
and so, perhaps, is the original nebular material transformed 
into stars such as men see clustered upon the sky of night. 

Certainly the force of gravitation must be active. Since 
the time of Newton, in the seventeenth century, it has been 
known that there is a force of gravitation ; that under the 
influence of that force every particle of matter in the uni- 
verse attracts or pulls every other particle of matter ; that 
the combined effect is always motion of some sort, each 
particle pursuing in space some determinate path or orbit 
under the influence of gravitational attraction exerted by 
all the particles. 

The most recent observations of nebulae have brought out 

4 





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PLATE 2. Spiral Nebula. 



THE UNIVERSE 

the fact that they are extremely numerous ; probably many 
hundreds of thousands exist, although only about ten thou- 
sand have been catalogued. This fact is of importance ; 
for if we are to regard the stars as a product of development 
or evolution from the nebulae, we should expect these 
gaseous bodies to exist in numbers comparable with the 
number of the stars themselves. 

But of even greater interest is another recent observation 
of the nebulae. The predominant type seems to be spiral 
in form; a species of central hub, carrying two attached 
curved spires, like a whirling wheel with two very flexible 
spokes but no rim. There can be little doubt that these 
nebulae are subject to internal motions, probably rapid in 
themselves, but appearing infinitely slow to us because of 
the almost inconceivably vast distance by which we are 
sundered from them. 

According to the foregoing theory, which admits the exist- 
ence of irregular as well as spiral nebulae, we should expect 
to find the stars in groups, a certain number assembled 
comparatively close together near certain former nebulous 
regions within the sidereal universe. And this is precisely 
what we do find. Usually the number of stars thus belong- 
ing together is small ; very frequently but a single star can 
be detected with the telescope. But many of the constit- 
uent stars of a group may be too faint to show themselves 
on account of their distance ; often they are all probably 
too faint, except the large one that may have resulted from 
the former hub or center of the parent nebula, if it was one 
of true spiral form. Furthermore, gravitational attractions 
and orbital motions may have commenced among the stars 
of every group even before they had become separate bodies. 
While they were still numerous, frequent collisions must 

5 



ASTRONOMY 

have brought about the coalescence of two or more into a 
single larger unit. In short, we have here an outline of a 
fairly consistent explanation to carry us forward from the 
nebular stage to that of stellar development, a theory that 
leads us to expect star-groups ranging all the way from a 
single visible luminous object to a detached assemblage 
closely packed in a globular cluster. 

And the stars are simply suns ; our sun is a star. There 
can be no doubt that the stars are not the tiny twinkling 
points of light they seem to be. Their apparent lack of 
size or volume is simply a result of the distance by which 
they are separated from us ; their twinkling comes from 
undulations or other irregularities in the ocean of terrestrial 
atmosphere, or "air," through which we are compelled to 
view them. We know the stars to be self-luminous, masses 
of glowing incandescent solids, liquids, or gases. We know 
the stars to be composed of chemical elements practically 
identical with those found on the earth. We know the stars 
to be subject to the law of gravitation ; that every particle 
of matter in each one of them is endowed with that mys- 
terious quality postulated by Newton, the power of pulling 
all other matter in the universe. From all these known 
facts, and reasoning by analogy, we are led to believe the 
stars to be suns, more or less like our own sun, though by 
no means necessarily in the same stage of cosmic develop- 
ment. All are doubtless cooling gradually and steadily by 
the constant radiation of heat into space ; some have 
probably reached temperature conditions similar to those 
existing in our sun ; and there may very probably be some 
that are attended by planets like our earth. 

The stars are classified according to their so-called magni- 
tudes, by which astronomers mean simply their lucidity or 

6 



THE UNIVERSE 

brightness, not their actual dimensions ; the first-magnitude 
stars are the brightest, the fifth-magnitude stars the faintest 
usually visible to the unaided eye under ordinary conditions. 
There are in all about sixteen hundred stars of the first five 
magnitudes ; and only about one half of these can be seen 
at any one time, when the sky is perfectly cloudless, because 
the other half are always concealed from view below the 
horizon. The stars are also divided into a series of so-called 
constellations ; very irregular, even grotesque imaginary 
figures of men, animals, and other objects, placed in the sky 
by the astronomers of old, and retained there in a somewhat 
simplified form by the moderns, principally on account of 
an unwillingness to destroy the ancient landmarks of this 
venerable and venerated science. 

The stars so far described are called fixed stars, which means 
that they do not change their relative positions in space ; 
that any two of them now close together have been thus in 
proximity from the beginning, and will remain so to the end. 
But modern researches have brought out the fact that these 
apparently fixed stars are not really fixed absolutely. They 
have motions in space ; these motions seem to us extremely 
slow and minute simply because the stellar distances are so 
vast. For at a sufficiently great distance, even large and 
rapid motions will necessarily appear reduced and retarded. 
And it is, in fact, quite inconsistent with what we know of the 
laws governing gravitational attraction to suppose any par- 
ticle of matter in the universe to be really fixed in position 
absolutely. Everything must move; must be following 
some duly appointed path, ever contrasting the intricate com- 
plexity of nature with the wondrous simplicity of nature's 
order and nature's law. Even our sun, regarded as a star, 
cannot be fixed in space, but must be moving majestically 

7 



ASTRONOMY 

through the void, drawing with it our attendant earth, and 
ourselves upon it. 

And if the stars are incandescent suns, we must expect to 
find, and we do find, that some among them undergo inter- 
nal changes that make their visible brightness vary. In 
certain cases slowly, in others more rapidly, their luminosity 
waxes and wanes with a more or less periodical regularity. 
Now and again, rarely and at long intervals, some special 
catastrophe takes place ; some convulsion of nature, whereby 
a new star is made to blaze forth into view where previously 
had been only darkness. Possibly we witness in such cases 
the result of a sudden collision in space between two ancient 
suns previously cooled through the ages, and long since 
bereft of luminosity and of life. The stars that change their 
brilliancy are called variable stars ; those that blaze forth 
suddenly are "new stars," or novce. 

As we have already stated, the sky contains stellar sys- 
tems other than those involving but a single visible object. 
Of these probably the most interesting are the double stars, 
composed of two individuals, often of different colors. These 
double stars appear but single to the unaided eye; only 
when the powers of a telescope of some size are brought into 
play, is it possible to resolve them into their component 
parts. In the field of view of such an instrument the stars 
all appear as brilliant points of light, occasionally glittering 
and sparkling, but the glitter and sparkle are imperfections 
caused by terrestrial atmospheric effects, and by the impos- 
sibility of constructing telescope lenses whose surfaces are 
ground to the right theoretic shape with absolute exactness. 
In other words, the stars appear in the telescope much as 
they do to the eye : only when the star is a double, the 
telescope often shows it as such, while the eye is unable to 

8 



THE UNIVERSE 

see between the two components. And it is a very impres- 
sive sight, when we turn a telescope upon one of these double 
stars, to see the two tiny points of light projected on the 
deep, fathomless background of the night sky, and to 
realize that the speck of darkness between them is a bit of 
abysmal space. 

Sometimes the close proximity of the components of a 
double star is fortuitous merely. The two objects may 
simply appear close together through happening to lie in 
almost exactly the same direction from us. But one of them 
may in reality be behind the other, and at a distance from 
us immeasurably greater than the first. In this respect 
astronomic observation differs from the viewing of ordinary 
objects on the earth. If, for instance, we should happen to 
notice two men, both standing at points almost exactly 
north of us, but one ten times as far away as the other, we 
would at once detect a difference of distance from the fact 
that the distant man would appear much smaller than the 
near one. But in the case of the stars, which we see as 
points of light merely, we could gather no such information. 
Even if one of the stars should be brighter than the other, 
this extra brilliancy might be due to a higher intrinsic light- 
giving power, and in no sense a result of greater proximity. 

When two stars thus appear close together, though in 
reality separated by a great distance, they probably have 
nothing in common, and are of lesser interest. But in 
certain cases the two stars will appear close together through 
really being near each other in space. Then they must 
belong to a single system; have probably originated in a 
single nebula ; true twin suns, bound one to the other and 
the other to the one ; held by the invisible, intangible, but 
indestructible power of gravitational attraction. 

9 



ASTRONOMY 

In addition to these fixed stars, whose motions were un- 
known to the ancients ; whose motions are so slow that 
generations of men must come and go before they can reveal 
themselves to the unaided eye, — in addition to these fixed 
stars, the night sky contains five other bright stars called of 
old the planets, from the Greek word irkavrJTrjs, the wanderer. 
They have been named Mercury, Venus, Mars, Jupiter, and 
Saturn. The most conspicuous thing about them, when 
viewed without a telescope, is their peculiar and rapid 
motion among the fixed stars. They can be seen to make 
an entire circuit of the heavens, traveling apparently among 
the fixed stars, in brief periods of time ranging from about 
a year to about thirty years. Of course we now know the 
cause. These planets are not properly stars at all; they 
are like the earth, attendants of our sun, revolving around 
the sun in perfectly definite paths or orbits, and in perfectly 
definite periods of time. Compared with the fixed stars, 
they are all extremely near the sun. And being all thus com- 
paratively near the sun, they are of course also all compara- 
tively near each other ; and our earth being one of the num- 
ber, they are all comparatively near the earth, too. But we 
have just seen that the extreme apparent slowness of stellar 
motion is really only a result of the extraordinarily great 
distance by which we are separated from the stars ; as this 
immensity of distance does not exist in the case of the 
planets, of course their apparent motions must and do 
appear to us comparatively rapid. 

Their apparent motions are also complex in a high degree. 
Two of them, Mercury and Venus, move around the sun in 
orbits smaller than that of the earth, and therefore entirely 
within the earth's orbit; the other three, Mars, Jupiter, 
and Saturn, are exterior to the earth. Mercury has the 

10 



THE UNIVERSE 

smallest orbit of all. It is always actually quite close to the 
sun, and therefore always appears near the sun when seen 
projected on the sky. Of course, it cannot be seen when the 
sun is visible on account of the overwhelming luminosity of 
the sun itself. Therefore we can observe Mercury occa- 
sionally only, just after sunset, near the point of the horizon 
where the sun has disappeared ; or just before sunrise, near 
the point of the horizon where the sun is about to make its 
appearance. It is thus always seen in the evening or morn- 
ing twilight, and was called of old the evening star or the 
morning star. The same is true of the planet Venus, which 
attains, however, a much greater apparent distance from 
the sun. 

The exterior planets, Mars, Jupiter, and Saturn, may be 
seen at certain times throughout a wide range of space on 
the sky, and at any hour of the night, all of which phe- 
nomena will be explained in detail in a later chapter. Still 
other planets exist ; but they are mostly too faint for the 
unaided eye; they have been discovered telescopically in 
modern times. All, together with our sun itself, are proba- 
bly the result of gradual changes in a parent nebula. 

The planets are unlike the stars in still another important 
particular. We have seen that the stars are self-luminous, 
incandescent ; the planets are quite different, and give 
out no light of their own. They shine only by reflected 
light which they receive from the sun. The light goes from 
the sun to the planet ; illumines it ; and then we see the planet 
by solar light, just as we see objects in a room by reflected 
solar light, which we call daylight. This produces a rather 
curious telescopic planetary phenomenon called phase, a 
phenomenon which is most conspicuous also in the case 
of our moon. The planets are globular in shape, and 

11 



ASTRONOMY 

therefore only one hemisphere can be illumined by the 
sun at any one time. But the planet does not usually 
happen to turn its illuminated hemisphere directly towards 
the earth. Therefore we usually see only a part of the bright 
hemisphere, and this often looks more or less like what is 
called a half-moon. In other words, we always see a hemi- 
sphere of the globular planet, but it is not the same hemi- 
sphere which is turned toward the sun, and which is therefore 
bright. If the hemisphere we see and the bright hemi- 
sphere are mutually exclusive, we see a dark or " new-moon" 
phase. If the bright hemisphere and the one we see over- 
lap, we see a crescent, half-moon, or other phase, as the case 
may be. Among the planets, Mercury, Venus, and Mars 
show the most conspicuous phase phenomena. 

Sir John Herschel has given a good illustration of dimer - 
sions in our solar and planetary system. Represent th3 
sun by a globe two feet in diameter. Then Mercury will 
be a grain of mustard seed on a circle 164 feet in diameter 
with the sun near its center ; Venus, a pea, 284 feet distant ; 
the earth, also a pea, 430 feet away; Mars, a pin's head, 
654 feet ; Jupiter and Saturn, oranges, distant respectively 
half a mile and four-fifths of a mile. The nearest fixed 
star, on the same scale, would be distant about 8000 miles, 
not feet. This illustration brings out clearly the compara- 
tively minute dimensions of the solar system in relation to 
the vastness of stellar distances. 

In actual appearance the planets differ greatly in the tele- 
scope; and they differ especially from the fixed stars. 
For even our most powerful optical apparatus will not 
suffice to magnify the latter so as to make them appear 
otherwise than as minute points of light. Many of them 
doubtless possess globular dimensions greatly exceeding any- 

12 



THE UNIVERSE 

thing we find in the solar system; but the vast distances 
cause these dimensions to shrink into mere nothings 
even in our largest telescopes. 

But in the case of the planets these great distances d 
not exist, and therefore the telescope shows their spheric? ( 
size in the plainest possible way. But the planets difTf 
greatly one from the other. Jupiter shows a bright, near! 
round disk, crossed by a few dark straight lines or.ba 
It is accompanied, in small telescopes, with four sate! 
or moons, which can be seen to revolve around the plane 
At times they pass behind the planet and disappear; and 
again, one or other of them is so placed that the planet inter- 
poses between it and the sun. Then, too, it disappears ; 
for the satellites also shine by reflected solar light ; and, of 
course, they receive none when Jupiter is placed between 
them and the sun. Finally, at certain other times a satellite 
may pass between Jupiter and the sun ; and then it can be 
seen to cast a small round shadow dot on the bright surface 
of the planet. Such phenomena are called eclipses. 

Saturn is the most beautiful of the planets, viewed with 
a telescope of moderate size. It has a number of moons or 
satellites, mostly too small to be seen in a glass of low power ; 
but its most conspicuous feature is the famous ring of Saturn. 
This is a flat disk surrounding the planet, and, in the words 
of Huygens, who was the first to explain it correctly, nowhere 
"sticking to" the planet. The ring, like the other bodies of 
our system, shines by reflected solar light ; and it is always 
distorted in appearance, as seen from the earth, into a flat- 
tened oval or ellipse, like a cart-wheel seen nearly edgewise. 
At certain times we actually do see it exactly edgewise, and 
then it appears, of course, like a thin, straight, bright line 
against the dark sky background. And when the ring 

13 



ASTRONOMY 

appears opened up to a considerable extent, we can see this 
dark background of the sky by looking through the openings 
between the ring and the ball of the planet. 

Tvlars and Venus show us plain bright disks of moderate 
size, exhibiting in small telescopes little or no detail of any 
kind in the way of markings or bands. Their most conspic- 
uous feature is the phase, which is much more marked than 
j is in the case of Jupiter and Saturn, whose phase 
henomena are practically altogether unnoticeable. This 
ollows, of course, from the fact that the quantity of visible 
phase is due to proximity ; and Mars and Venus, being the 
planets nearest to our earth, must, of course, show more 
phase than the distant planets Jupiter and Saturn. 

Mercury, as we know, is seen only in the twilight, showing 
in the telescope a small disk with marked phases. 

Comets are occasional visitors to the solar system. They 
come presumably from outer space in the course" of their 
orbital motions under the influence of gravitational and 
perhaps other forces ; remain for a time in the vicinity 
of the solar system ; are consequently visible to us ; and 
finally retire again into the depths of space whence they 
came. When bright enough to be observed without the tele- 
scope, they commonly exhibit to our view a brilliant come- 
tary "head," containing a central condensation or nucleus 
surrounded by a mass of tenuous luminous haze, and to it 
often attached a long visible streamer or tail, in olden times 
dreaded by all as a possible harbinger of wars and pestilence. 

All these cometary phenomena are well seen in the 
photograph reproduced as a frontispiece in the present 
volume. The tail in this case has more than one streamer ; 
and its length, as photographed, is about 11°, or nearly 
one-eighth the distance from the horizon to the zenith, 

14 



THE UNIVERSE 

The tail actually seen by astronomers was at one time 
twice as long. The little curved lines on the photograph 
are star-images. We should of course expect these to be 
round dots in the picture ; but in photographs of this kind 
they are almost always drawn out into little curves, for a 
very simple reason. The telescope is aimed accurately at 
the comet when the exposure of the photographic plate is 
commenced, and it is kept thus pointed at the comet during 
the whole duration of the exposure. This of course makes 
a "moved picture" of the stars, as photographers would 
call it. For the comet will " wander " among the stars, like 
a planet, in consequence of its orbital motion in space ; and 
if the telescope's movement upon its stand is adjusted cor- 
rectly to allow for the comet's motion, the photographic 
images of the stars must suffer. 

The earth, considered as an astronomic body, is but one 
of the smaller planets ; yet in one respect it is the most 
important of all, since it is the one upon which we live. 
Astronomers have been able to ascertain many facts about 
the earth, which we shall for the present summarize with 
the utmost brevity, postponing all detailed description to a 
later chapter. We know, first, that the earth rotates once 
daily on an axis ; that this rotation carries us around, 
too ; that in consequence of it the sun, stars, and other 
heavenly bodies seem to rise in the east, climb upward 
in the sky, and finally sink down again and set in 
the west. We also know that our earth, like the other 
planets, travels around the sun in an orbit ; that it requires 
a whole year to complete a circuit of that orbit ; that in 
consequence of the daily axial rotation and the yearly orbital 
revolution, we experience the phenomena of night and day, 
summer and winter, — phenomena to be explained fully 

15 



ASTRONOMY 

later ; finally, we know from actual measures made upon the 
surface of our planet that the earth is a slightly flattened 
globe about 8000 miles in diameter. 

The moon is the only satellite of our earth, and by far the 
most conspicuous object in the night sky; the most beauti- 
ful and interesting of all the heavenly bodies when observed 
through small telescopes ; and important especially as being 
our nearest neighbor in the whole wide domain of cosmic 
space. Again summarizing existing knowledge as briefly 
as possible, the moon is now thought by astronomers to 
have once formed a part of the earth ; to have been set 
free in some very distant age in the past by the action in some 
way of gravitational and possibly other forces. It revolves 
around the earth in an orbit somewhat similar to the earth's 
own annual orbit around the sun ; completes such an orbital 
revolution in about twenty-seven and one-quarter days ; 
and, in consequence thereof, appears to make a complete 
circuit among the far more distant fixed stars and planets 
in the same period, traveling around from a position of 
apparent proximity to any given fixed star back to the 
same star again in the twenty-seven and one-quarter day 
period. It is not self-luminous or incandescent, but shines 
by reflected sunlight like the planets ; in consequence of its 
nearness to the earth, it exhibits the most pronounced phase 
phenomena, varying all the way from the full-moon, down 
through the half-moon stage, to actual invisibility at the 
time of new-moon. It is about 240,000 miles distant from 
the earth ; is about 2000 miles in diameter ; and the gravi- 
tational attraction of its mass upon the waters of terrestrial 
oceans gives rise to the ebb and flow of the tides. 

The physical or actual appearance of the moon is not 
unlike that of the earth. The surface, as seen in the tele- 

16 







. 




A 








¥ ^^f^k f :J- ', 


. 


J 




• 




Brat ' ' 






CL ■ 



Photo at Lick Observatory. 

PLATE 3. The Moon in the First Quarter Phase. 



THE UNIVERSE 

scope, is very much broken; there are several mountain 
ranges, and, especially prominent, a great number of large 
craters, apparently of volcanic origin, and usually having 
a mountain peak in the center. Extremely conspicuous 
features of the lunar surface, as seen in small telescopes, are 
the very black shadows which are cast on the surface when 
sunlight falls obliquely on the mountains and craters. 
There is no air or other atmosphere, and no water; nor 
have we any reliable evidence that any perceptible changes 
have taken place in the volcanic surface features since accu- 
rate records of telescopic observation were begun by men. 

To complete this preliminary brief outline survey of our 
subject, it remains to add a few words about the sun, the 
central body of our solar system. The sun is our source of 
light and heat ; without it life, as we know it, would be im- 
possible on our earth. It is about ninety-three million miles 
distant from us, and nearly a million miles in diameter ; 
within its vast bulk might be placed the earth and moon, 
together with the entire lunar orbit in which, as we have said, 
the moon revolves around the earth in twenty-seven and one- 
quarter days. The sun turns on an axis in a period of about 
twenty-five terrestrial days ; its surface is usually marked 
by the well-known sun spots, visible in small telescopes 
plainly, and first seen by Galileo, when he turned upon the 
sun probably the first telescope ever made. These spots 
are now known to have periods of special frequency. Every 
eleven years they occur in greater numbers than usual; 
and this period of eleven years is in some mysterious way 
connected with the known frequency periods of auroras 
and magnetic storms on our earth. The bulk and mass of 
the sun are so great that its gravitational attraction far 
exceeds that of all the planets combined. It thus becomes 

c 17 



ASTRONOMY 

the gravitational ruler of the whole solar system ; around it 
all the planets may be said to revolve in their duly appointed 
paths or orbits. 

It is hoped that the foregoing brief summary of astronomic 
science may help to awaken a desire in the reader to possess 
more detailed knowledge; and this we shall endeavor to 
give in later chapters ; perhaps we may be permitted to 
conclude the present one by calling attention to the value of 
astronomy for practical purposes as well as for mental dis- 
cipline and study. It is often said that astronomy is a 
somewhat detached subject ; of interest certainly, but hav- 
ing little or no close and intimate relation to the everyday 
affairs of human life. But in reality the converse is the 
truth. Probably no other of the more abstruse sciences 
enters so directly and so frequently into our daily affairs 
as does astronomy. There are at least three services per- 
formed by astronomy that are essential, and without which 
civilization, as we know it, would be impossible. These 
things are : first, the regulation of time ; second, the exe- 
cution of boundary surveys and the making of maps and 
charts ; third, navigation. 

Few persons stop to think when they enter a jeweler's 
shop to correct their watches by comparison with the jeweler's 
" regulator," or when they communicate by telephone with 
a central telephone station to ask for the correct time, 
that both the jeweler and the telephone operator must 
themselves have some source of correct time by which to 
regulate their regulators. This source of correct time is 
the astronomical observatory. The standard observatory 
clock is itself but a fallible piece of machinery fabricated by 
fallible human hands, and it can be kept right only by con- 
stant comparisons, made on every clear night, with the 

18 



THE UNIVERSE 

unvarying time standards provided by nature, the stars 
themselves in their courses. For instance, time observations 
of the stars are made regularly and nightly in the United 
States Naval Observatory at Washington, the chief official 
astronomic station of the United States government. With 
these observations the standard clocks in the clock room of 
the observatory are corrected and timed ; and from these 
standard clocks electric signals are sent out daily in accord- 
ance with a pre-arranged schedule so that time-balls can 
be made to indicate the exact instant of noon to the people, 
and jewelers and others may correct their regulators. 
Thus is every citizen in touch with the astronomic observatory 
almost daily and of necessity, although he does not generally 
realize the fact until it is brought specially to his attention. 

And the matter of mapping and charting is equally depend- 
ent upon astronomy. Ordinary small surveys of farms or 
towns may be made by ordinary surveyor's instruments 
without constantly having recourse to astronomers. But 
of what value would be a map of an entire continent unless 
the customary latitude and longitude lines were inscribed 
upon it ? And these essential lines cannot be so inscribed 
without astronomic observations. Such observations must 
necessarily be made specially for the purposes of each survey, 
and the consequent calculations always depend, too, upon cer- 
tain prior astronomic data contained in published astronomic 
"tables" or printed books, themselves in turn based on aver- 
age or mean results obtained in the great observatories of the 
world during the last couple of centuries by steady con- 
tinuous systematic study and observation of the stars. 

Even more important than continental maps for the 
progress of civilization are the coast charts published by the 
various governments of the maritime nations. These also 

19 



ASTRONOMY 

require very precise latitude and longitude lines ; and here, 
as before, recourse must be had to astronomic observations 
and accumulated astronomic results. 

Finally, navigation itself, upon the open sea, could not 
proceed successfully without astronomy. Those of our 
readers who have crossed the ocean in a magnificent modern 
steamer may have seen at times the captain or navigating 
officer "take the sun," as it is called, with a sextant. Pos- 
sibly they have thought that after making such an obser- 
vation the navigator could read on the face of the sextant 
the exact position of the ship at the moment, its latitude and 
its longitude on the earth, as ordinarily understood in 
geography. But such is by no means the fact. Before 
they can be made to yield this essential information, 
sextant observations must be subjected to a somewhat 
laborious process of numerical calculation, or " reduction," 
as it is called. This is an astronomic process ; and in carry- 
ing it to completion the navigator again requires certain 
printed tables of a purely astronomic character. These 
are contained in a book called the " nautical almanac," 
which is published annually in various languages by the 
several civilized governments of the world. And again, as 
before, for the preparation of such nautical almanacs, these 
governments must maintain, and do maintain, astronomic 
computing bureaus, manned by astronomers, and employing 
in their calculations once more the published results obtained 
by astronomers of the past in the various great fixed obser- 
vatories. The details of all these astronomic activities 
must, of course, be postponed to later chapters ; but it is 
hoped that enough has been said here to remove from the 
reader's mind the possible notion that astronomy is of little 
or no practical utility in the ordinary affairs of men. 

20 



THE UNIVERSE 

But far beyond and above all this, the study of astronomy 
possesses a value peculiarly its own, as a means of mental 
training. On account of venerable age and consequent 
approximate perfection of knowledge, this science is char- 
acterized especially above all others by the peculiar intri- 
cacy of the elementary problems it presents, and by the 
unusual exactness of which their solutions admit. Further- 
more, notwithstanding the importance of its direct practical 
applications, which have been mentioned, the study of astron- 
omy is peculiarly free from any materialistic tendency, — 
from any connection, in short, with utilitarian motives. 
It is not a vocational study, giving knowledge which can be 
sold for money by the young college graduate upon his entry 
into practical affairs. But it is preeminently a study which 
will give a clearer outlook upon the universe in which we 
pass our lives, preeminently one that will make that universe 
seem a pleasanter place in which to live. So that if a certain 
portion of our time is to be devoted to studies that are 
not strictly vocational, astronomy will surely be found a 
profitable and desirable subject. And surely also there is 
much to be gained in our choice of studies from the selection 
of such as are likely to arouse a real interest in the student ; 
to arouse that desire for knowledge which, once awakened, 
will make the task of the teacher an easy one. Here again 
astronomy holds a most favorable place. That which has 
its being within the confines of a single drop of water is as 
wonderful as are the motions within a planetary or sidereal 
system. But the animalcules within that drop of water, 
though their number be myriad, can never stir our deepest 
interest, for they are without that strong appeal to the imag- 
ination, without those vast distances and mighty forces, the 
materials of astronomic study alone. 

21 



CHAPTER II 

THE HEAVENS 

Probably the best method of approaching the study of 
astronomy is to begin with those observations and problems 
that do not require the use of any instruments whatever. 
These problems are surely the earliest problems, since 
men of old must have begun to discuss the mysterious 
events they could see about them in the universe long before 
they had invented even the rudest instruments of measure- 
ment. 

Astronomy is a study of the sky ; and the first thing to be 
noticed in a study of the sky is the sky itself. To us it ap- 
pears at night like a great, round, blue, hollow dome within 
which we are standing. To its interior surface seem to be 
attached the apparently numberless bright twinkling points 
of light we call stars. In the day it carries only the sun, 
and perhaps, too, the moon rather faintly visible; and in 
the intermediate periods which we call twilight, and which 
occur at dawn and at dusk, we can see perhaps two or three 
dim stars, called morning and evening stars. We know 
that these morning and evening stars are certain of the 
planets, which, as we have already seen, are members of 
the solar system like our earth, circling around the sun, 
each in its proper path or orbit. 

But there is no real dome of the sky above and around us ; 
it is simply an optical illusion, a creation of our own imagina- 
tion. Nevertheless, it is most convenient to imagine it to 

22 



THE HEAVENS 

be real, because we can thus fix our first astronomical 
ideas to something tangible ; and by a consideration of this 
round dome as if it actually existed, we shall be able to 
clarify and to solve many interesting problems. Granting, 
then, that there is such a dome above us, we have no reason 
to imagine it other than perfectly round. Let us regard it 
as a great hollow ball or sphere; astronomers have given 
it the name Celestial Sphere. 

The next question is whether this celestial sphere is the 
same sphere everywhere. Is the celestial sphere surround- 
ing New York identical with that surrounding the city 
of Capetown, South Africa ? The answer is : yes. The 
sphere is the same sphere everywhere. Theoretically, the 
center of the sphere is at the center of the earth ; and since 
the diameter of the earth is about eight thousand miles, an 
observer on the earth's surface will be distant about four 
thousand miles from the true center of the sphere. But 
such a distance as four thousand miles is literally a mere 
nothing compared with the infinitely vast distance of the 
celestial sphere. The whole planet earth shrinks into a 
mere dot in comparison. It makes absolutely no difference 
whether you are on the earth's surface, or could be transferred 
to its center, you would see identically the same imaginary 
celestial sphere. The stars and other heavenly bodies, 
wherever they may be situated around us in space, seem to 
be projected upon that distant celestial sphere, and attached 
to its interior surface. Even if you could make a sudden 
jump of about ninety- three million miles from the earth to 
the sun, you would still see the same identical sphere, much 
too far away to be affected by such a little change in the 
observer's position. Not only the earth, but its entire 
orbit, including the sun, shrink into a dot. Astronomy is 

23 



ASTRONOMY 



truly a science of vast distances. But there is this essential 
difference between the distance of the celestial sphere 
and all other distances in the science. The far-ness (if we 
may use such a word) of this imaginary sky sphere is in- 
finitely greater than any other actually known and meas- 
ured by men. 

The accompanying Fig. 1 is intended to illustrate this 
notion of the celestial sphere. The large circle is supposed to 

represent the sphere ; 
only, of course, its size 
cannot be made big 
enough ; the reader 
must imagine it ex- 
tended to infinity. 
The dot E at the cen- 
ter of the big circle is 
the earth; the reader 
and the author are sup- 
posed to be standing 
on the surface of that 
dot. The tiny circle 
represents the earth's 
annual path around 
the sun, the sun itself being the larger dot at the center of 
the tiny circle. The crosses represent stars scattered through 
sidereal space at all sorts of distances from the earth. The 
lines with arrows passing through the crosses indicate the 
points on the interior surface of the celestial sphere where 
the stars will appear to be projected, and where they will 
seem to be attached to the interior or supposedly visible 
surface of the sphere. The longest arrow indicates the point 
on the sphere where the sun will appear projected, that arrow 

24 




Fig. 1. The Celestial Sphere. 



THE HEAVENS 

being, of course, merely a straight line passing from the earth 
to the sun and thence continued outward to the sphere. 
For the sun will also appear to us as if attached to the 
interior surface of the sphere, like the stars, at the point 
indicated by its arrow. This elementary notion, that the 
various celestial bodies will appear to be located on the 
sphere at the points shown by their arrows is an important 
idea, and one that is not at all difficult to grasp. We must 
not forget that the arrows are all supposed to be infinitely 
long ; even the solar arrow is infinite, although the sun dot 
and the earth dot are very near each other, cosmically 
speaking. 

Having thus fixed our ideas as to the celestial sphere, 
we must next study it in its relation to the various objects 
that appear projected upon it ; and the first important 
thing to consider more in detail is the position of the sun on 
the sphere. We have already seen that the earth travels 
around the sun once in a year. The path or orbit in which 
the earth thus travels is an oval or ellipse ; but for the 
purpose of a first approximation such as we shall here con- 
sider, we can take this path of our earth to 
be a circle, with the sun at its center. 
Now this circular orbit, like every circle, 
must lie entirely in a single plane or flat 
surface. The accompanying Fig. 2 shows 
this circular approximate orbit of the earth 
E moving around the sun at the center S FlG - 2 - The Earth's 
in the direction shown by the arrow. The 
single plane or flat surface in which the entire orbital path 
lies is here of course the flat plane of the paper on which 
this page is printed. 

It is evident that the earth, being always in its orbit, 

25 




ASTRONOMY 

must likewise always be situated in the plane of the paper. 
And the sun, being at the center of the circular orbit, must 
also be in the same plane. From these considerations fol- 
lows the important preliminary principle that earth and sun 
are both constantly in a single plane. To this important 
fundamental plane has been given the name Plane of the 
Ecliptic. 

The plane of the ecliptic is defined, then, as the plane in 
which are situated at all times the sun, the earth, and the 
earth's orbit around the sun. Now let us extend our ideas 
so as to include the celestial sphere in our consideration of 
the earth's orbit. Imagine the orbital plane, but not the 
orbit, extended or stretched outward, indefinitely, farther 
and farther, approaching gradually an infinite bigness, until 
at last it meets the imaginary celestial sphere. Evidently, 
it will cut out a circle on the celestial sphere, just as though 
one were to slice a round orange with a flat cut. The line 
in which the rind of the orange would be severed by such a 
cut would then be a circular line ; and so also must the 
line cut out on the celestial sphere by the ecliptic plane be a 
circle. The fact that the sphere is an immense globe and 
the orange a small ball here makes no difference. The 
principle is the same. 

It is possible to draw a little more information from the 
analogy of the orange. Wherever we slice the orange, we 
obtain a circle ; but if it was sliced through the center, the 
orange would be cut in two equal halves, and then the 
circle would be the largest circle that could possibly be 
drawn around the rind of the orange. Applying this to the 
case of the celestial sphere cut by the ecliptic plane, we see 
at once that here also the sphere is cut in two equal halves. 
For the earth, as we have seen, is at the center of the celes- 

26 



THE HEAVENS 

tial sphere ; and therefore the ecliptic plane, which passes 
through the earth, is also a cut or slice through the center 
of the celestial sphere. Consequently, the circle cut out 
on the celestial sphere by the ecliptic plane produced to 
infinity is a circle as large as can possibly be drawn on the 
celestial sphere, and it divides that sphere in two equal halves. 
Such a circle drawn on a sphere, dividing it into halves, is 
called a Great Circle of the sphere. The particular great 
circle of the celestial sphere, cut out by the plane of the 
ecliptic produced to infinity, is called simply the Ecliptic. 

The ecliptic, then, is defined as a great circle of the celestial 
sphere cut out by the plane of the earth's orbit around the 
sun, produced to infinity. It would be a convenience if 
some one could go up to the sky and mark out the ecliptic 
circle upon it with a big paint-brush. While this is im- 
possible, it is perfectly easy to mark it upon a celestial globe ; 
and the reader is advised to examine such a globe, when he 
will surely find the ecliptic plainly drawn upon it. 

The important peculiarity of the ecliptic circle is this : 
the sun must always at all times appear to lie in that circle. 
And the reason is quite simple, as shown again in Fig. 3. 
Here we have once more drawn a large circle to represent 
the infinite celestial sphere ; and the dot which should 
represent the combined sun, earth, and earth's orbit around 
the sun is shown at the center, magnified into a circle. The 
observant reader will notice, upon comparing Figs. 1 and 3, 
that in the former figure the earth occupies the center of 
the sphere, whereas in Fig. 3 the sun is at the center. But 
the figures are interchangeable, as we already know, because 
of our having assigned infinite size to the celestial sphere. 

In Fig. 3 the smaller circle represents the earth's orbit 
around the sun, E f and E" being two positions of the earth 

27 



ASTRONOMY 

in its orbit. The corresponding apparent positions of the 
sun, as projected on the celestial sphere, are shown at S' 
and S". For, as we already know, if an imaginary line be 
drawn from the earth to the sun, we must necessarily see 



Celesfia/Sc. 




Fig. 3. The Ecliptic Circle. 

the sun from the earth along the direction of that imaginary 
line ; and if the line be extended outward until it pierces 
the celestial sphere, the sun will appear to us projected on 
the sphere at the point where the sphere is pierced by the line. 
Now this sight line from the earth to the sun will neces- 
sarily lie entirely in the plane of the earth's orbit, for in that 
plane both the earth and the sun are at all times situated. 
Consequently, the sight line, when extended to pierce the 
celestial sphere, must necessarily always pierce that sphere 
somewhere on the circle cut out on the sphere by the plane 

28 



THE HEAVENS 

of the earth's orbit produced outward to infinity. But 
this circle is the ecliptic ; and thus we have a proof that the 
sun must always appear on the sky projected upon the 
ecliptic circle. And it is certainly a most remarkable thing 
that it should thus be possible to draw an imaginary circle 
on the sky such that at all hours of the day, on every day 
of the year, and of every year, when we look at the sun, it 
will appear to be situated at some point of that circle. Yet 
it all follows quite simply from the above elementary con- 
siderations concerning our earth's orbital motion around 
the sun. And it is furthermore already equally evident 
that as the earth progresses around its orbit, as shown by 
the curved arrow, the sun will appear to progress around the 
ecliptic circle with a rate of motion corresponding to the 
earth's own motion in its orbit. 

Figure 3 also gives a good opportunity to explain the 
meaning of the terms "angle" and " angular distance," 
which we shall have frequent occasion to use. An angle is 
defined as the difference in direction between two lines. 
Thus, if we consider the lines SS' and SS" in Fig. 3, the angle 
between them is indicated by the combination of letters 
S'SS". Every angle is thus indicated by a combination of 
three letters ; the middle letter of the three always indicating 
the point of the angle, or its so-called " vertex." The cor- 
responding angular distance on the celestial sphere between 
S' and S" is the arc S'S" ; and such angular distances must 
of course be measured in degrees. In Fig. 3 the angular 
distance S'S" is about 120°, or one-third of an entire circum- 
ference of 360°. 

These facts about the ecliptic constitute one of the most 
important discoveries of the very earliest astronomers. The 
hazy records of extreme antiquity indicate that the Chinese 

29 



ASTRONOMY 

knew the ecliptic and had measured its position on the sky 
as early as 1100 b.c. The early Greek astronomers of Alex- 
andria certainly knew of it ; for instance, we have fairly 
reliable records showing that Eratosthenes (276-196 b.c.) 
measured its position quite accurately. 

The next important phenomenon to which our attention 
must be directed results from still another motion of our 
earth ; namely, its axial rotation. As we all know, the earth 
turns on its axis once daily ; a motion which is quite distinct 
from its orbital revolution around the sun. Both motions 
take place simultaneously, the earth traveling around the 
sun in its orbit while it is at the same time spinning on its 
axis, much as a couple of waltzing dancers move from end to 
end of the room while at the same time spinning rapidly 
around each other. 

This terrestrial rotation has an immediate effect upon the 
celestial sphere and all the heavenly bodies which appear 
projected upon it. For the astronomer, being fastened to 
the earth, turns around with it, perforce. And as the earth 
turns, with the astronomer attached, it is constantly pre- 
senting him to a new part of the celestial sphere. Just so 
a dancing couple face every point of the compass in suc- 
cession, in consequence of their spinning motion, and quite 
independent of the fact that they are also moving about 
in the room at the same time. 

This turning of the astronomer successively toward dif- 
ferent parts of the celestial sphere makes that sphere appear 
to him as though it were turning around the earth instead of 
the earth turning within it, precisely as a railway passenger 
sees fields and trees apparently flying past his train, although 
he knows these objects are really fixed in position, and him- 
self in rapid motion. 

30 



THE HEAVENS 

The axial rotation of the earth takes place from west to 
east ; and the consequent seeming rotation of the celestial 
sphere is from east to west. Objects projected on the 
sphere partake of this seeming motion ; the sun, the moon, 
the morning and evening stars, and all the other stars. 

This is the cause of day and night ; of the rising and setting 
of all heavenly bodies, including the sun. As the earth 
rotates from west to east, they all seem to revolve in the 
opposite direction daily, rising from beneath the eastern 
horizon, slowly climbing the sky, and again sinking down 
to set in the west. These facts are quite generally known 
with respect to the sun and moon, but comparatively few 
are aware that the stars also rise and set. It is reported 
that Sir George Airy, a recent astronomer royal of England, 
used to say that not more than one person in a thousand 
knows that the stars, like the sun, rise and set. Most 
people think the stars are always the same, simply a uniform 
countless assemblage of thickly clustered luminous points. 

Having thus explained the earth's rotation, we must next 
consider its rotation axis. Our planet earth, in its rotation, 
turns about an imaginary line or axis passing through its 
center and meeting the earth's surface at the north and south 
poles of the earth. Now imagine for a moment this rotation 
axis extended outward in both directions, farther and farther, 
until at last the two ends pierce the celestial sphere itself. 
They would, of course, mark out on the sphere two points 
corresponding exactly to the two terrestrial poles. These 
two points are called the north and south Poles of the 
heavens, or the celestial poles. The long line joining them 
is the axis of the celestial sphere, and a very short bit near 
the middle of the line is the terrestrial rotation axis. Figure 
4 again shows the celestial sphere, this time with the earth 



ASTRONOMY 



CeJestisL^^ 




at the center, magnified from its proper size of a mere dot, 
so as to exhibit the earth's rotation axis and its prolongation, 

the axis of the celes- 
tial sphere. N and S 
are the north and 
south poles of the 
earth; NS is the 
terrestrial rotation 
axis ; and its pro- 
longation to the ce- 
lestial sphere marks 
out N' and S', the 
north and south 
poles of the celestial 
sphere. 

Now since the ap- 
parent rotation of the 
celestial sphere is merely a result of the earth's turning, 
and since the latter takes place around the axis, so also the 
great sphere's seeming turning must take place about this 
same axis. In other words, all the stars must seem to re- 
volve nightly around the two poles of the heavens. Stars 
very near the poles on the sky will seem to turn in little 
circles ; those farther from the pole will seem to turn in 
larger and larger concentric circles. 

Figure 5 shows a few of these circles in which the stars 
appear to revolve nightly, and indicates that those near 
the two poles of the heavens are small. As we go farther 
from the poles the circles become larger, until at last we 
come to stars halfway between the two celestial poles, 
where the largest of all the circles occurs. The circles are 
of course all parallel ; and they are concentric in the sense 

32 



Fig. 4. The Celestial Poles. 




THE HEAVENS 

that their real centers all lie on a single straight line, the axis 

of the celestial sphere. The stars, as they appear to revolve 

in the circles, of course 

complete a revolution 

every twenty-four 

hours, since the axial 

rotation of the earth 

within the sphere is 

the true cause of the 

whole phenomenon ; 

and this axial rotation 

occupies exactly one 

day of twenty-four 

hours. And because 

of this daily period, 

the circles are called 

Fig. 5. Diurnal Circles. 

Diurnal Circles of the 

celestial sphere. Diurnal circles are defined, then, as par- 
allel circles on the celestial sphere in which the stars com- 
plete their daily apparent rotation around the celestial poles. 
We have just seen that the largest of all the diurnal 
circles is the one halfway between the two celestial poles; 
and it is a particularly important one. It of course divides 
the entire celestial sphere in two halves, which are called 
the northern and southern celestial hemispheres, and this 
largest diurnal circle is itself called the Celestial Equator. It 
corresponds exactly to the equator on the earth, which 
similarly divides our planet into northern and southern 
hemispheres. In fact, it is clear that as the terrestrial and 
celestial poles correspond exactly, so also the terrestrial 
and celestial equators must correspond exactly. And it is 
therefore also possible to define the celestial equator in a 
d 33 



ASTRONOMY 

manner quite analogous to the definition of the other impor- 
tant great circle of the celestial sphere, the ecliptic. For 
if, as in the case of the ecliptic plane, we imagine the plane 
of the earth's equator stretched out and extended until it 
finally reaches the celestial sphere, it will cut out a great 
circle on the sphere, and this great circle is the celestial 
equator. So we might define the celestial equator as a 
great circle on the celestial sphere cut out by the plane of the 
earth's equator produced to infinity, and this definition 
is equivalent to the former one, which describes the celestial 
equator simply as the largest of all the diurnal circles. 

Having thus defined the celestial poles and equator, 
it is easy to carry analogy a little farther, and inquire what 
corresponds on the sky to latitude and longitude on the 
earth. The reader will recall from geography that when we 
desire to define the position of a place on the earth we do so 
by giving its latitude and longitude. Terrestrial latitude 
is defined as the angular distance of a place north or south 
of the earth's equator, and terrestrial longitude is its angular 
distance east or west from some so-called " prime meridian," 
such as that of Greenwich, England. 

Exactly analogous methods are used for defining a star's 
place on the sky, or the location of the point where it ap- 
pears to us projected on the celestial sphere. Unfortunately, 
the terms celestial latitude and longitude have not been 
used for this purpose. Instead of these terms, astronomers 
use the words " declination " and " right-ascension " ; which 
bear the same signification with respect to the celestial 
equator that terrestrial latitude and longitude bear to the 
equator on our earth. 

It is of interest to consider here the initial point from which 
astronomers reckon right-ascensions ; for there is no prime 

34 



THE HEAVENS 



meridian on the sky like that of Greenwich on the earth. 
Instead, astronomers use an initial point on the celestial 
equator, and from it the right-ascensions of all celestial 
objects are counted. This point is called the Vernal Equi- 
nox, and its location will be understood easily from the fol- 
lowing considerations. 

We have so far defined two great circles of the celestial 
sphere, each dividing the sphere in two halves. They are 
the ecliptic circle and the celestial equator. Now these 
two circles, as shown 
in Fig. 6, must inter- 
sect at two opposite 
points on the sphere ; 
for any pair of great 
circles on any sphere 
must evidently do 
this. These two op- 
posite points are 
called equinoctial 
points ; one is the 
Vernal Equinox, the 
other the Autumnal 
Equinox. We shall 
have occasion farther 
on to explain the im- 
portance of these two points a little more in detail ; for our 
present purpose we need merely remember that the vernal 
equinox point is by universal convention selected as the 
initial point for measuring all right-ascensions. 1 

At the risk of seeming somewhat tiresome, we must still 
add to these rather prolix preliminary explanations a very 

x Note 1, Appendix. 
35 




Fig. 6 



Two Great Circles intersecting at Opposite 
Points of the Sky. 
(After Cassini's Astronomie, p. 78. Paris, 1740.) 



ASTRONOMY 

few more necessary definitions. For there is still another 
important great circle on the celestial sphere, again dividing 
it into a pair of halves, but a different pair from the two 
hemispheres north and south of the celestial equator. 
This important great circle is the Horizon. In astronomy 
the horizon is precisely the same thing as the horizon in 
ordinary life. It is defined accurately as a great circle on 
the celestial sphere cut out by an infinitely extended level 
plane touching the earth at the point where the observer 
stands. 1 Of course, in the interest of exactness, we should 
note in passing that the same horizon circle would be cut 
out on the sky by a plane parallel to the first, but passing 
through the earth's center beneath the observer's feet. 
This is, of course, again a result of the fact that the earth's 
radius of four thousand miles, by which distance these two 
planes are separated, is a perfectly negligible quantity in 
comparison with the infinite distance of the celestial sphere. 

Having defined the horizon, it is easy to add two other 
definitions, both of which refer to astronomical terms having 
also the same signification precisely that they bear in ordi- 
nary English. These are the Zenith, which is simply the 
point of the celestial sphere directly overhead, and therefore 
exactly 90° distant from every part of the horizon; and 
Altitude, or angular elevation above the horizon. Altitude 
is defined accurately thus : the altitude of a celestial body is 
its angular distance (p. 29) above the horizon. The altitude 
of the zenith is thus evidently 90°. 

As we now know the meaning of the two points on the 
celestial sphere called the celestial north pole and the zenith, 
it is possible to define next the Celestial Meridian. This is 
a great circle drawn on the sphere from the celestial north pole 

1 This plane, in mathematical language, is a plane tangent to the earth. 

36 



THE HEAVENS 



to the zenith, and thence extended completely around the 
sphere until it returns again to the pole. Very simple con- 
siderations show that the celestial meridian must pass 
through the north and south points of the horizon. 1 

The accompanying Fig. 7, representing a celestial globe, 
may make the foregoing description clearer. The circle 
HVO is generally 
made of wood, and 
represents the celes- 
tial horizon. HPZAO 
is usually made of 
brass, and represents 
the celestial meridian, 
passing through the 
celestial pole P, the 
zenith Z, the north 
point of the horizon 
H, and the south point 
of the horizon O. The 
circle ASQ is the celes- 
tial equator, every- 
where 90° distant from 
the pole P. The cir- 
cle BC is a diurnal circle. ZV is a flexible strip of brass 
marked with degrees and pivoted at Z. It can be turned 
to any part of the horizon, and, by means of the degree 
divisions marked upon it, we can measure the altitude or 
angular elevation of any star above the horizon. Some of 
the constellation figures (p. 7) are also drawn on the globe. 

Having now defined the principal circles and points upon 
the celestial sphere, let us next investigate the position of the 

1 For additional definitions and explanations, see Note 2, Appendix. 

37 




Fig. 7. The Celestial Globe. 
(From Lalande's Astronomie, 3 ed., Tome 1, p. 74. 
1792.) 



Paris, 



ASTRONOMY 




north pole of the sphere with respect to our horizon. We 
shall first imagine an observer standing at the north pole 

of the earth. It is 
p' 
n vg> 1 -~-^ evident from the ac- 

companying Fig. 8 
that such an ob- 
server would see the 
celestial pole directly 
overhead, in the 
zenith. For P being 
the observer's posi- 
tion at the north pole 
of the earth, PS will 
be the earth's rota- 
tion axis, passing 
through the two 
poles of the earth. 
And if this axis is 
lengthened out to an infinite size, it will meet the celestial 
sphere at P', the north pole of the sphere, which will clearly 
be directly overhead. PH is a level plane touching the 
earth where the observer stands ; consequently H is a point 
of the horizon, in accordance with our definition (p. 36). 
OH f is a plane passing through the earth's center parallel to 
the level plane PH ; and the points H and H' will coincide on 
the celestial sphere because the distance PO is absolutely neg- 
ligible in comparison with the infinite distance of the sphere. 
These considerations show that to an observer at the pole of 
the earth the celestial pole will be at the zenith, and its alti- 
tude, or angular elevation above the horizon, will be 90°. 

To an observer standing at the terrestrial equator the 
position of the pole will be quite different, as shown in Fig. 9. 

38 



Fig. 8. 



Observer at the North Pole : 
in the Zenith. 



Celestial Pole 



THE HEAVENS 




If we place the observer on the earth at E, and call E a 
point of the equator, the terrestrial rotation axis will be at 
PS, because any 
point on the terres- 
trial equator must 
be 90° distant from 
the terrestrial poles. 
This puts the celes- 
tial north pole at 
P', which coincides 
with K, a point on 
the horizon of an ob- 
server at E. It fol- 
lows from this that 
if we go to the equa- 
tor of our earth, we 
will there see the ce- 
lestial pole in our 
horizon, distant 90° from our zenith at Z, directly overhead. 
Having thus ascertained that to an observer at the pole 
of the earth the celestial pole appears overhead, and to one 
at the equator in the horizon, it is not difficult to realize 
that an observer traveling from the pole to the equator 
will see his celestial pole gradually seem to move down 
from his zenith to his horizon. For if the celestial pole 
occupies two extreme positions in the zenith and horizon 
when the observer is in two extreme terrestrial positions 
at the pole and equator, it is clear that as the observer 
occupies successive intermediate terrestrial positions, the 
celestial pole will seem to occupy successive positions also, 
intermediate between the zenith and horizon. This is 
the reason why travelers going south, and noting the pole 

39 



Fig. 9. 



Observer at the Equator ; Celestial Pole in 
the Horizon. 



ASTRONOMY 



star night after night, see that star gradually sinking lower 
in the sky; and if they continue southward quite to the 
equator, they see the pole star actually disappearing at the 
horizon. For the pole star is so placed in space as to be 
projected on the sky very near the imaginary celestial pole ; 
and consequently the visible pole star partakes of the changes 
which we have just explained. 1 In fact, the altitude, or 
angular elevation of the celestial pole above the horizon, is 
everywhere equal to the observer's terrestrial latitude, or 
angular distance from the terrestrial equator. 

This very important theorem enables us at once to study 
the very different appearance of the celestial sphere and its 

diurnal circles as seen 
from different places 
on the earth. At the 
equator, where the 
pole is in the horizon, 
the celestial sphere 
looks like Fig. 10, 
called the Right 
Sphere. Here the 
diurnal circles (p. 33) 
are all perpendicular 
to the horizon, and they are all bisected or halved by the 
horizon. Consequently, as the celestial bodies perform their 
daily apparent rotation with the sphere, in consequence of 
the corresponding daily axial rotation of the earth inside, — 
their diurnal circles being all halved by the horizon, — all 
the celestial bodies will be above the horizon just as long 
as they are below it. They will be "up" twelve hours, and 
"down" (or "set") twelve hours. 

x Note 3, Appendix. 
40 




Fig. 10. The Right Sphere. 
(After Long's Astronomy, Vol. 1, p. 91. Cambridge, 1742.) 



THE HEAVENS 

Now we have seen (p. 29) that as the earth travels around 
the sun in its annual orbit, the sun seems to travel around 
the ecliptic in a corresponding manner. But, wherever it 
may be projected on the ecliptic, it must always be on a 
diurnal circle ; in the light of what we have just learned about 
the right sphere, this diurnal circle must be halved by the 
horizon; therefore, to an observer at the equator, the sun 
will be above the horizon twelve hours every day in the 
year. We therefore see that at the equator day and night 
are always equal throughout the whole year. 

Quite a different state of things holds at the pole, where 
we see what is called the Parallel Sphere, as indicated in 
Fig. 11. Here the ce- 
lestial pole is at the 
zenith, and the diur- 
nal circles are all par- 
allel to the horizon. 
If a celestial body is 
above the horizon at V 

\ Horizon 

all, its entire diurnal 
circle is above the 

horizon ; it Will re- FlG - n - The Parallel Sphere. 

(C j j i ■ (After Long's Astronomy.) 

mam up twenty- 
four hours during each axial rotation of the earth. The 
largest diurnal circle, the equator, here coincides with the 
horizon ; to an observer in the northern hemisphere, stars 
between the celestial equator and the north pole never 
set; those between the equator and the south pole never 
rise. 

How would these facts affect the sun, which is always 
seen in the ecliptic, as we know? We also know that the 
ecliptic is halved or bisected by the celestial equator (p. 35). 

41 





ASTRONOMY 



Therefore, during half the year the sun will be between the 
equator and the north pole. During that half-year its 
successive diurnal circles on the parallel sphere will be 
entirely above the horizon, and the sun will not set. This 
explains the important and well-known fact that at the 
north pole the sun remains above the horizon six months, 
and day, as well as night, is six months long. 

To observers situated on the earth in places like New 
York, intermediate between the pole and the equator, the 

sky appears in the 
form called the Ob- 
lique Sphere, shown 
in Fig. 12. Here the 
diurnal circles are 
neither perpendicular 
to the horizon, nor 
parallel to it. Being 
parallel to each other, 
they all make the 
same angle with the 
horizon, an angle which is different in different terrestrial 
latitudes. 

And the diurnal circles are not halved by the horizon, 
either. Each such circle is divided by the horizon in two 
unequal parts. If the circle is between the celestial equator 
and the north celestial pole, as B, Fig. 12, the part above the 
horizon is the longer. If the circle is between the equator 
and the south pole, as E, the part below the horizon is the 
longer. Thus it follows that stars projected on the sky 
between the equator and the north celestial pole are above 
the horizon each day longer than they are below it, and 
vice versa. Only stars on the celestial equator itself have a 

42 




Fig. 12. The Oblique Sphere. 
(After Long's Astronomy.) 



THE HEAVENS 

halved diurnal circle, and are above and below the horizon 
equal twelve-hour periods. Some of the diurnal circles 
quite near the north pole, as A, do not reach the horizon 
at all. Stars projected on these diurnal circles will there- 
fore never set ; and stars with corresponding diurnal circles 
near the south pole will never rise. 1 Observers in the south- 
ern hemisphere of the earth, of course, have these conditions 
reversed. 

The sun, always projected on the ecliptic, may have its 
diurnal circle divided either way. We have seen (Fig. 6, 
p. 35) that the ecliptic is bisected or halved by the equator. 
Consequently, when the sun is seen in one half of the ecliptic, 
it is between the equator and the pole, and therefore above 
the horizon longer than below it ; and when it is seen in the 
other half of the ecliptic, it is below the horizon longer than 
above it. In the one case, the days are longer than the 
nights ; in the other, the nights are longer than the days. 
As the sun is seen in one half of the ecliptic during about 
half of each year, it follows that during half of each year 
our days are longer than our nights in the temperate regions 
of the earth, where the oblique sphere prevails. Only when 
the sun is exactly on the equator, at one of the two points 
where it is intersected by the ecliptic, does the sun have a 
halved diurnal circle, giving us equal periods of light and 
darkness, — equal days and nights. We have already seen 
that these two points of intersection of the equator and 
ecliptic are called equinox points. We now know the 
origin of the name ; when the sun is seen projected at either 
of these points of the ecliptic, we have equal days and nights. 
It may facilitate the comprehension of these facts if the 
reader will again examine Fig. 7, p. 37, the Celestial Globe. 

1 Note 4, Appendix. 
43 



ASTRONOMY 

From the above elementary considerations follows at 
once a preliminary understanding of the phenomenon called 
the Seasons. For it is clear that the half-year during which 
our days are longer than our nights will be a summer or 
hot half-year, since we obtain our heat from the sun; and 
the half-year with the long nights will be a cold or winter 
half-year. Near the terrestrial equator, where the right 
sphere gives constantly equal days and nights, there must 
be, and is, a complete uniformity of seasons. Near the 
pole, with its parallel sphere, there is a long six months' 
summer day, and a corresponding winter night. But the 
polar summer is itself cold, because even in summer the sun 
never rises to a great altitude above the polar horizon. 



44 



CHAPTER III 

HOW TO KNOW THE STARS 

Any one beginning the study of astronomy quite naturally 
desires to proceed as quickly as possible from the reading 
of books about the stars to an examination of the stars 
themselves in the sky. And in a first preliminary survey 
it is of interest to learn the names of the principal stars and 
constellations as they have been handed down to us from 
olden times. It is not at all difficult to acquire this knowl- 
edge, now that we have become acquainted (Chapter II) 
with the celestial sphere, and the more important lines and 
circles which astronomers imagine to be drawn upon that 
sphere. 

There are four objects that often puzzle beginners, when 
they attempt to compare star maps with the night sky for 
the purpose of identifying the more important bright stars. 
These are the same four things that so puzzled the ancients, 
the four bright planets, Venus, Mars, Jupiter, and Saturn. 
Mercury, the only other bright planet, is rarely seen ; but 
one or more of the above four is almost sure to be conspicu- 
ous in the sky, to a certain extent impairing the correctness 
of our star maps. 

For these star maps do not show the planets ; and for a 
very simple reason. We know (p. 10) that the planets seem 
to wander among the fixed stars ; they appear, now here, 
now there, at very widely varying points of the sky. On 
the other hand, the fixed stars (p. 7) retain relative posi- 

45 



ASTRONOMY 

tions practically unchanging; if, for instance, any three 
are located in a straight line, they will continue to lie on that 
straight line for centuries, so far as observations with the 
unaided eye can ascertain. It would be possible to make a 
star map showing both stars and planets as they appear 
on any given date. But such a map would not be correct 
six months later; while it would still show the fixed stars 
in their proper relative positions, the planets would be 
wrongly placed, on account of their wanderings. For this 
reason, astronomers omit the planets altogether from their 
star charts ; and beginners are puzzled. 

For the beginner, upon looking at the sky, always ob- 
serves the planets first of all, because they appear as bright as, 
or brighter than, the most brilliant fixed stars on account of 
their proximity to the earth. For instance, the beginner 
may see three lucid stars forming a small triangle, with the 
brightest star of the three at the apex of the triangle. He 
at once looks at the star map, to identify this triangle. 
Finding none in the proper place, he always concludes that 
he has misunderstood the printed directions ; packs up his 
books and lantern, and returns indoors, discouraged. The 
beginner in astronomy is always modest as to his abilities, 
and blames himself if the universe fails to fit the printed 
directions. Nor does any real astronomer ever lose this 
modest characteristic of the beginner; for he who has 
studied this science most deeply is ever most of all convinced 
that he is still a beginner. 

Of course the absence of the triangle from the star map 
was simply due to the extremely brilliant object at the 
apex being a planet, and therefore properly absent from the 
map. The triangle on the sky appears on the map as a 
simple straight line with but two stars upon it. 

46 



HOW TO KNOW THE STARS 

Therefore the beginner should first of all learn to know the 
planets, so that he can eliminate them in comparing his star 
map with the sky. And it is fortunately easy to become 
familiar with the planets, perhaps even easier than to learn 
the stars. We have merely to take advantage of the planets' 
superior brilliancy in order to identify them. The best 
way is to make observations in the dusk, after sunset, before 
the stars begin to become visible. If there is any bright 
planet above the horizon at that time, it will be the first 
to show itself in the twilight ; it will be the evening star. 

But this priority of appearance in the evening is not nec- 
essarily a sure test for distinguishing the planets; for if 
no planet is above the horizon at the moment of sunset, the 
first object seen in the dusk of the twilight sky will, of course, 
be the brightest of the fixed stars then above the horizon. 
Therefore it is important to have another criterion for 
identifying the planets. It is a fact that the planets always 
appear projected on the sky rather near the ecliptic circle 
(p. 27). Therefore, if we could locate the position of the 
ecliptic circle on the celestial sphere, we should have addi- 
tional evidence as to whether the evening star appearing 
first is really a planet. If a planet, it must be near the 
ecliptic circle. 

The following method will enable the beginner to locate 
the ecliptic circle approximately on the sky. One point 
of the circle is, of course, determined by the position of the 
sun, which, as we know (p. 27), is always seen projected 
on that circle. Consequently, as we are making these ob- 
servations in the evening twilight, it follows that one point 
of the ecliptic is near that point of the horizon where the 
sun has just set. 

If we can now locate on the sky one other point of the 

47 



ASTRONOMY 

ecliptic, we can determine roughly the location of the entire 
ecliptic circle ; for two points are sufficient to locate any 
great circle on the sky. This can be done best by mak- 
ing use of the celestial meridian, which we recall as a great 
circle drawn on the sky from the zenith directly overhead 
down to the south point of the horizon (p. 36). The point 
of the meridian crossed by the ecliptic can be ascertained 
from the following little table, which gives the roughly 
approximate altitude, or angular elevation above the hori- 
zon, of that point on the meridian which is crossed by the 
ecliptic. 

To use this table, it is merely necessary to face the south 
point of the horizon, and imagine the meridian drawn ver- 
tically upward on the sky from that point to the zenith 
overhead. Next we must imagine the entire distance on 
the meridian from horizon to zenith divided into ninety 
equal degrees or spaces. Then the table gives us for various 
terrestrial latitudes, and for various dates, the number of 
degrees between the horizon and the point of the meridian 
at which the ecliptic crosses it. To facilitate the practical 
use of the table we have placed in it, next to each number of 
degrees, a simple fraction which will perhaps be more con- 
venient in making actual observations. Thus, where the 
table gives 46°, we find also the fraction J, meaning that the 
ecliptic crosses the meridian approximately halfway up 
from the south point of the horizon to the zenith. The 
fraction \ belongs with 46°, because 46 is approximately 
half of 90°, the total angular distance from horizon to zenith. 

For example, if we should observe at New York (approxi- 
mate latitude 40°) at sunset on January 1, we would imagine 
a great circle drawn around the sky from the sunset point of 
the horizon to a point on the meridian halfway between 

48 



HOW TO KNOW THE STARS 



Table for Finding the Ecliptic at Sunset 

Angular Altitude of its Intersection with the Meridian above the South 
Point of the Horizon 





Latitude 30° 


Latitude 40° 


Latitude 50° 


January 1 . . 


58° f I 


46° ) « 


32° i« 


February 1 . . 


73 H 


62 |« 


49 f.« 


March 1 . . . 


82 f" 


71 tS 


61 IS 


April 1 . . . 


83 fj 


73 f* 


63 IS 


May 1 . . . 


78 f* 


66 f* 


54 *j 


June 1 . . . 


75 ff 


52 f? 


39 ** 


July 1 . . . 


52 If 


40 ff 


26 ff 


August 1 . . 


42 ff 


31 If 


20 if 


September 1 


38 ff 


28 if 


18 if 


October 1 . . 


37 f? 


27 f? 


17 i? 


November 1 


39 f? 


28 | w 


18 ir 


December 1 . . 


45 J? 


34 f J 


22 i« 



the zenith and the south point of the horizon. This imag- 
ined line would be part of the ecliptic; extending it beyond 
the meridian, and around the sky to the eastern horizon, 
would give us the remaining visible portion of the ecliptic. 
And any object suspected of being a bright planet would 
necessarily be found very near this ecliptic circle. If the 
moon should chance to be visible at sunset, it would give us 
an additional point near the ecliptic; for the moon likewise 
always appears in the immediate vicinity of that circle. 

Actual observations of this kind will of course be extended 
in the twilight for about an hour after sunset. As the posi- 
tion of the ecliptic changes somewhat during that hour, 
we have added two letters to each number in the table. 
One of the letters is either a u or a d, and shows whether 
the ecliptic point on the meridian is moving up or down at 
the moment of sunset. The other letter is either an n or 

e 49 



ASTRONOMY 

an s, and indicates whether the ecliptic point on the horizon 
is moving north or south at the moment of sunset. Thus, 
an hour or more after sunset on January 1 at New York (lati- 
tude 40°), we should draw the ecliptic a little to the north 
of the observed sunset point in the horizon, and a little 
above the 46° point on the meridian. 

If we find a brilliant object in the dusk in this way on the 
ecliptic, we may still further test its planetary character 
by the absence of twinkling, for planets do not twinkle as 
much as stars. If the suspected object shines quietly, 
serenely, almost without scintillation, we may be tolerably 
sure it is a planet. 

Still another important aid is at the service of the begin- 
ner in his planetary search, — the ordinary almanac. This 
will tell him what " evening stars" or planets are visible 
on the date when he makes his observations; and it is 
certainly a great help to know in advance whether any 
planets are to be in sight. The almanac will also inform him 
as to the names of the planets he may expect to see. 

But even without an almanac it is generally easy to dis- 
tinguish between the different planets. Mercury, when 
visible, always appears very near the horizon, close to the 
point where the sun has set. The best date to look for it 
may be found by adding successive periods of one hundred 
and sixteen days to the initial date, Nov. 2, 1913. The 
planet can usually be seen for a few days before and after 
the dates obtained in this way, if the horizon is unusually 
free from cloud or mist. Conditions are most favorable 
when the computed dates occur in the early part of the 
year, from January to May. And in these months espe- 
cially it is important to begin looking for Mercury at least 
a week before the predicted dates. 

50 



HOW TO KNOW THE STARS 

Venus should be sought after sunset on the ecliptic ; its 
angular distance from the sun is never more than 47° (about 
one-quarter of a great semicircle of the sky) ; and it may be 
much less. In looking for it, about an hour after sunset, 
we must remember that in an hour the sun will have moved 
a considerable distance below the horizon; therefore, 
even if Venus is 47° distant from the sun, we must expect 
its distance from the sunset point of the horizon to be con- 
siderably less. An initial date when Venus attains its 
greatest distance from the sun is Feb. 12, 1913. Subse- 
quent occurrences of the same phenomenon may be expected 
at intervals of 584 days thereafter (1.60 years). These 
dates are, of course, highly favorable for observing the 
planet. Both Mercury and Venus are extremely bright. 

Mars, Jupiter, and Saturn also always appear near the 
ecliptic, but they may attain very great angular distances 
from the sun. They are, in fact, directly opposite the sun 
in the sky at certain dates, which are the most favorable 
dates for finding these planets. The dates are : 

Mars, Jan. 5, 1914, and thereafter at intervals of 780 days 
Jupiter, July 5, 1913, and thereafter at intervals of 399 days 
Saturn, Dec. 7, 1913, and thereafter at intervals of 378 days 

When thus opposite the sun, the planets are easily found. 
It is merely necessary to imagine a straight line drawn from 
the sun to the observer, and thence continued outward to 
the celestial sphere at a point opposite the sun. And if 
we imagine the line drawn an hour after sunset, we must 
not draw it from the sunset point of the horizon, but from 
the sun itself, making an approximate allowance for the 
sun's having moved some distance below the horizon during 
the interval of an hour since sunset. On these critical dates 

51 



I 



ASTRONOMY 

Mars, Jupiter, and Saturn are on the meridian, due south, 
at midnight. 

For some time after the critical dates, these three planets, 
always remaining near the ecliptic, diminish their angular 
distances from the sun at the approximate monthly average 
rate of 25° for Mars, 30° for Jupiter, and 32° for Saturn. 
In estimating such angular distances it is well to remember 
that the angular diameter of the full moon is about one-half 
a degree. Furthermore, all the above numbers vary some- 
what in different years. The interval of 780 days between 
successive critical dates for Mars is especially variable : it 
is usually only about 750 days when the predicted date 
occurs in the early months of the year. 

A final test as to the planets may be obtained if the ob- 
server has a small telescope or good field glass at his disposal. 
In such an instrument the planets show their round disks 
quite plainly, while the fixed stars appear in the field of 
view as mere points of light without any visible extension 
into disks. In a three-inch telescope Jupiter shows moons, 
usually four, and Saturn usually exhibits the ring. Most 
observers detect in Mars a sort of reddish or ruddy color. 

Coming now to the identification of the fixed stars, we 
shall employ a method resembling somewhat our procedure 
in the case of the planets. It is not our purpose to include 
in the present volume detailed charts showing all stars 
visible to the unaided eye, but rather to confine our atten- 
tion to the stars of especial brilliance, and the more con- 
spicuous constellations with which every one should have 
an acquaintance. 

The first things to find in the sky are the pole star and 
the constellation Ursa Major (Great Bear or " Dipper"). 
These objects are near the north celestial pole, and very 

52 



HOW TO KNOW THE STARS 



far from the ecliptic ; consequently, the planets never appear 
among them to confuse the visible configurations of stars. 
The pole star, close to the north celestial pole, is always 
elevated above our horizon by an angular altitude very 
nearly equal to the observer's latitude (p. 40). To find it, 
we must therefore face the north, and imagine the celestial 
meridian drawn on the sky vertically upward from the north 
point of the horizon to the zenith. The pole star will then 
be found almost exactly on the meridian, and elevated above 
the horizon by an angle equal to the observer's terrestrial 
latitude. In New 
York, for instance, 
it will be elevated 
41°, or about f of the 
total angular dis- 
tance from horizon 
to zenith. The pole 
star is not very bril- 
liant ; being of the 
second magnitude, it 
will be inferior to sev- 
eral of the brightest 
stars visible in vari- 
ous parts of the sky. 

To verify this fig. 13. 
identification of the 
pole star we make use of Ursa Major. This constellation con- 
tains seven stars, not of the first magnitude, arranged as shown 
in Fig. 13. This figure exhibits the constellation as it appears 
in the sky at 9 p.m. about April 21 in each year. The reader 
will notice that the two end stars of the seven are in the me- 
ridian directly above the pole star, and that they point almost 

53 




The Pole Star and Ursa Major as seen at 
9 p.m. on April 21. 



ASTRONOMY 

exactly toward the pole star. For this reason these two 
stars are called "The Pointers." If these seven stars appear 
on the sky occupying the position shown in Fig. 13 with 
respect to the pole star at 9 p.m. about April 21, there is 
no doubt that the pole star has been identified correctly. 
In using Fig. 13, the reader should bear in mind that the 
constellation Ursa Major will appear much larger on the 
sky than it does in the figure. The scale of the figure has 
been so chosen that the distance of Ursa Major from the 
pole star is proportioned correctly to the elevation of the 
pole star above the horizon ; and this choice of scale makes 
the constellation appear rather small. The other con- 
stellation figures, 14, 15, 17, 18, 19, 20, 21, 22, are all drawn 
to the same scale, to avoid confusion ; and the reader must 
expect all these constellations to be larger on the sky than 
they appear in the figures. 

In consequence of the seeming rotation of the celestial 
sphere about the pole (p. 32), the pointers will further occupy 
the positions shown in Fig. 14, at 9 p.m. on the several dates 
indicated in the figure. 

On intermediate dates the pointers will of course occupy 
intermediate positions ; and with the help of these figures 
the reader should have no difficulty in finding the pole star 
and making certain of its identification by means of the 
pointers. 

There is one other interesting constellation near the 
celestial pole: Cassiopeia, the "Lady in the Chair." It is 
found easily, also, by the aid of the pointers. Imagine a 
straight line drawn from the pointers to the pole star, and 
continued beyond the pole star an angular distance equal 
to the distance between the pointers and the pole star. 
The end of the line will then be in Cassiopeia, and the appear- 

54 



HOW TO KNOW THE STARS 




55 




ASTRONOMY 

ance of that constellation is shown in Fig. 15. It looks like 
the letter W. The arrow shown in the figure indicates the 
direction of the pole star from Cassiopeia, and is approxi- 
mately a continuation of the line by means of which Cassio- 
peia was found. In comparing Fig. 15 
with the sky, it is therefore necessary to 
turn the book around until the arrow is 
nearly parallel to the direction of the 
pointers from the pole star. This would 
make the arrow vertical upwards, as shown 
in Fig. 15, at 9 p.m. on May 18, and verti- 
Cassiopeia ca ] downwards at 9 p.m. on November 18. 

May 18. „ 

It would be horizontal to the right on 
February 18, at 9 p.m. ; and horizontal to the left on August 
18, 9 p.m. On intermediate dates the arrow would of course 
occupy positions intermediate between these vertical and 
horizontal ones ; always, of course, at the hour of 9 p.m. 

Having thus indicated a method of finding the two impor- 
tant polar constellations, we shall next show how to identify 
the brightest fixed stars of the first magnitude visible in the 
United States and Europe. They are fifteen in number; 
in the following list we have arranged them in the order of 
luminosity, the brightest of all being placed first. 

To find these stars, we shall use a method similar to that 
employed for locating the ecliptic circle on the sky. Let 
the observer face the south at 9 p.m., and imagine the merid- 
ian drawn on the sky vertically upward from the south 
point of the horizon to the zenith, directly overhead. Let 
him once more imagine the meridian divided into ninety 
degrees or spaces, beginning at the south point of the hori- 
zon, and ending at the zenith. The following table will 
then tell him the dates on which the various stars in question 

56 



HOW TO KNOW THE STARS 

First-magnitude Stars 



Name 


Constellation 


Color 


Sirius 


Canis Major 


(Big Dog) 


Blue-white 


Vega 


Lyra 


(Harp) 


Blue-white 


Arcturus 


Bootes 


(Bear-keeper) 


Orange 


Capella 


Auriga 


(Charioteer) 


Yellow 


Rigel 


Orion 


(Hunter) 


White 


Procyon 


Canis Minor 


(Little Dog) 


White 


Betelgeuse 


Orion 


(Hunter) 


Red 


Altair 


Aquila 


(Eagle) 


Yellow 


Aldebaran 


Taurus 


(Bull) 


Red 


Antares 


Scorpius 


(Scorpion) 


Red 


PoUux 


Gemini 


(Twins) 


Orange 


Spica 


Virgo 


(Virgin) 


White 


Fomalhaut 


Piscis Australis (Southern Fish) 


Orange 


Regulus 


Leo 


(Lion) 


White 


Deneb 


Cygnus 


(Swan) 


White 



appear on the meridian, and their altitude or angular eleva- 
tion above the south point of the horizon when they are 
thus situated on the meridian, always at the hour of 9 p.m. 
The date of reaching the meridian at 9 p.m. is the same for 
all terrestrial latitudes ; but the altitudes vary in different 
latitudes, and are therefore given in the table for latitudes 
30°, 40°, and 50°. If the observer's latitude is intermediate 
between 30° and 40°, or between 40° and 50°, he can of course 
use altitudes intermediate between those given in the table. 
Sometimes the tabular altitudes are a little greater than 90°. 
This indicates that the stars in question cross the meridian 
north of the zenith. To see them, an observer facing south 
would need to bend his head back so as to see a little beyond 
his zenith. A better way is to turn around and face the 
north, when the stars in question will be seen very near the 
zenith. 

57 



ASTRONOMY 



The identification of the bright stars will, of course, in- 
clude an identification of the important constellations in 
which they are situated, as indicated in the preceding table. 

Table to be used in Finding First-magnitude Stars on the 
Meridian at 9 p.m. 





Date on Merid- 
ian, 9 P.M. 


Altitude above South Point 


of Horizon 




Lat. 30° 


Lat. 40° 


Lat. 50° 


Sirius . . . 


Feb. 15 


43° 


33° 


23° 


Vega . . . 


Aug. 15 


99 


89 


79 


Arcturus . . 


June 10 


80 


70 


60 


Capella . . 


Jan. 23 


106 


96 


86 


Rigel . . . 


Jan. 23 


52 


42 


32 


Procyon . . 


Mar. 1 


65 


55 


45 


Betelgeuse 


Feb. 2 


67 


57 


47 


Altair . . . 


Sept. 3 


69 


59 


49 


Aldebaran . . 


Jan. 13 


76 


66 


56 


Antares . . 


July 13 


34 


24 


14 


Pollux . . . 


Mar. 2 


88 


78 


68 


Spica . . . 


May 28 


49 


39 


29 


Fomalhaut 


Oct. 20 


30 


20 


10 


Regulus . . 


Apr. 8 


72 


62 


52 


Deneb . . . 


Sept. 16 


105 


95 


85 



The above table is correct at 8 p.m. instead of 9 p.m. on 
dates two weeks later than those given in the table. It 
is correct at 10 p.m. on dates two weeks earlier than the 
tabular dates. 

To facilitate finding the bright stars on dates other than 
those on which they reach the meridian at 9 p.m., we now 
give another table containing the dates when these stars 
rise and set at 9 p.m. as seen from the three terrestrial lati- 
tudes 30°, 40°, and 50°. In addition to the dates of rising 
and setting, the table contains the direction (as N.E., S.W., 

58 



HOW TO KNOW THE STARS 



WNW 




wsw 



ESE 



etc.), to which the observer must turn in order to see his 
star rise or set. In making these observations it is impor- 
tant to remember 
that the immediate 
vicinity of the ho- 
rizon is usually ob- 
structed by trees, 
houses, etc., and 
that even when 
these obstructions 
are absent, the ho- 
rizon itself is sel- 
dom entirely free 
from clouds or mist. 
Therefore the ob- 
server should not 
expect a rising star 
to be visible for 
some time (possibly as much as an hour) after 9 p.m. on the 
tabular date of rising ; and he may expect it to disappear 
from view some time before 9 p.m. on the tabular date of 
setting. 

The directions N.W., S.E., etc., to which the observer 
must turn, are roughly approximate only ; but accurate 
enough to facilitate finding the stars. The accompanying 
Fig. 16 shows the order in which these directions follow each 
other around the horizon. 

The table on the next page is correct at 8 p.m. instead of 
9 p.m. on dates two weeks later than those given in the table. 
It is correct at 10 p.m. on dates two weeks earlier than the 
tabular dates. 

To aid still further in the identification of the finest con- 

59 



Fig. 16. The " Points of the Compass. 



ASTRONOMY 



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AVNAV 

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53 


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flJ ft © 3 oct;^ &^o^ 3© & 


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May 7 
Dec. 12 
Sept. 21 
June 1 
Apr. 20 
June 3 
May 9 
Dec. 8 
Apr. 24 
Sept. 26 
June 20 
Aug. 21 
Dec. 30 
July 16 
Jan. 20 


3 
pi 

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g « § | clS, £§o 

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60 



HOW TO KNOW THE STARS 




Fig. 17. Auriga, with Capella. 




Fig. 18. Cygnus, with Deneb. 




Fig. 19. Gemini, with Pollux. 



ASTRONOMY 




Fig. 20. Leo, with Regulus. 




Fig. 21. Scorpius, with Antares. 




Fig. 22. Orion, with Rigel and Betelgeuse. 



62 



HOW TO KNOW THE STARS 

stellations, we have prepared the preceding diagrams exhibit- 
ing their appearance when rising, when setting, and on the 
meridian. In each case the diagram contains an arrow 
showing the direction of the pole star ; and the dates when 
the several constellations may be seen at 9 p.m. can be taken 
from the preceding tables. 

Those of our readers who may desire to extend their 
knowledge to the less conspicuous constellations may now 
do so easily. It is merely necessary to proceed from the 
constellations already known to those not yet identified, 
by the aid of a star atlas. In doing this it will be best to 
look for the known constellations and first-magnitude stars 
on the maps, and proceed from them first to the neighboring 
unknown constellations. There is little difficulty in doing 
this ; the knowledge of a few stars with which to begin is 
the only troublesome part of the problem. It is hoped that 
the tables and diagrams of the present chapter will suffice 
to remove this initial difficulty. 

It is also possible to identify the stars by means of a 
globe such as that illustrated in Fig. 7 (p. 37), but it is not 
easy to learn the method of using a globe without the aid 
of oral teaching. A few minutes' explanation from some 
person who understands the use of the instrument is better 
than many printed pages in a book. There is also another 
contrivance, called a planisphere, which is simple in use, and 
much less costly than a celestial globe. This instrument 
represents the globe projected on a plane or flat surface ; 
and by means of a rotating disk of cardboard, it shows at a 
glance what stars are visible above the horizon at any hour 
of the night and on any date in the year. Planispheres 
are always accompanied with printed instructions suitable 
for use by a beginner in astronomy. 

63 



ASTRONOMY 

In a study of the present chapter the reader will have 
noticed that we have given practical directions for finding 
the stars, without elaborate explanation of the principles 
upon which these directions are based. This will enable 
him to commence his study of the sky without waiting until 
he has mastered the later chapters of the book ; it is hoped 
to increase his interest by thus allowing him to undertake 
practical work at the earliest possible moment. 



64 



CHAPTER IV 

TIME 

We have seen (p. 19) that it is one of the principal duties 
of the astronomer so to regulate clocks that they may indi- 
cate accurate time : let us now endeavor to explain the 
meaning of the word "time" in astronomy. We shall 
make use of our definition (p. 36) of the celestial meridian as 
a great circle of the celestial sphere passing through 
the celestial pole, the zenith, and the north and south 
points of the horizon. For in astronomy, this meridian 
plays a most important part in the explanation of no less than 
four different kinds of time. These are called : — 

1. Sidereal time. 2. Apparent solar time. 

3. Mean solar time. 4. Standard time. 

A unit of some sort is necessary for measuring the dura- 
tion of these various varieties of time : and for this purpose 
astronomers use the Day; though not the same "day" for 
the four different kinds of time. There is a sidereal day, 
for measuring sidereal time; an apparent solar day, for 
apparent solar time; and a mean solar day, used for both 
mean solar and standard time. 

Let us consider first the simplest kind of time, sidereal or 
"star-time." We have had (p. 35) a definition of the vernal 
equinox as one of the points on the celestial sphere at which 
the ecliptic circle crosses the celestial equator ; and we have 
already made some use of this important point. We shall 
now find that it is fundamental also in the measurement 
of sidereal time. 

f 65 



ASTRONOMY 

As the celestial sphere performs its diurnal seeming 
rotation, due to the real axial turning of the earth within it, 
the vernal equinox, like the stars, rotates with the sphere. 1 
Consequently, once during each complete diurnal rotation 
of the sphere, the vernal equinox will cross the celestial 
meridian. At the precise instant when the vernal equinox 
thus crosses the celestial meridian, the sidereal day begins. 
As the seeming turning of the sphere proceeds from east to 
west, the vernal equinox will begin to move westward from 
the meridian as soon as the sidereal day has commenced; 
and after a complete rotation, it will again reach the meridian 
from the east. The sidereal day will then end, and at the 
same instant a new sidereal day will begin. The sidereal 
day is defined, then, as the interval of time between two 
successive returns of the vernal equinox to the meridian. 

The sidereal day is divided into twenty-four sidereal 
hours ; and these hours are counted continuously from 
to 24, without using the letters a.m. and p.m. When the 
vernal equinox is exactly on the meridian, and the sidereal 
day begins, the sidereal time is h m s ; and this would be 
the time indicated on the dial of a standard sidereal clock, 
if the clock were exactly right. Then, after the vernal 
equinox has passed the meridian, and has completed one 
twenty-fourth part of an entire diurnal rotation, it is l h m 
s sidereal time ; 2 h , 3 h , 4 h , etc., follow in succession ; 
until, at 23 h sidereal time, the vernal equinox lacks but one 
hour of reaching the meridian once more. 

When the vernal equinox is l h west of the meridian, we 
say that its " hour-angle" is l h ; and similarly for 2 h , 3 h , 
etc., up to 24 h . Thus the hour-angle of the vernal equinox 
at any moment may be defined as the quantity of rotation 

1 Cf. Note 2, Appendix. 
66 



TIME 

made by the celestial sphere since the vernal equinox was 
last on the meridian, this rotation being measured in hours, 
minutes, and seconds, and an entire rotation of the sphere 
corresponding to 24 hours. And in the light of this definition 
we may define the sidereal time at any instant as the hour- 
angle of the vernal equinox at that instant. 1 Recurring to 
our definition of right-ascension (p. 34), it may be here 
stated as an additional fact that the right-ascension of any 
star appearing on the celestial meridian at any instant is 
always exactly equal to the sidereal time at the same instant. 2 

This last important fact calls attention to a simple and 
interesting relation between sidereal or star-time, and the 
stars themselves. If, for instance, we have at hand a correct 
sidereal clock, and that clock indicates 3 h sidereal time 
exactly, then any star whose known right-ascension is 3 h may 
be found at that moment on the meridian. Furthermore, 
sidereal time enables us to know at once how much time has 
elapsed since any given star was on the meridian. Thus, 
at 4 h sidereal time, we know that our star, whose right- 
ascension is 3 h , passed the meridian one hour ago. At 
5 h we know it was on the meridian two hours ago, etc. ; 
and thus we know approximately where to look for it in the 
sky. 3 

We must next consider the explanation of solar time, and 
its relation to sidereal time. Let us begin with apparent 
solar time, which is the kind of time kept by the actual sun, 
as we see it in the sky. The definitions are quite similar 
to those we have already given for sidereal time. The unit for 
measuring the duration of apparent solar time, the apparent 
solar day, is defined as the interval between two successive 
returns of the visible sun to the celestial meridian. The 

1 Note 5, Appendix. 2 Note 6, Appendix. 3 Note 6, Appendix. 

67 



ASTRONOMY 

day begins when the sun is exactly on the meridian ; when 
the axial turning of the sphere has carried it one twenty- 
fourth part of an entire diurnal rotation westward from the 
meridian, astronomers say it is l h apparent solar time, etc. 
Following the analogy of sidereal time, we may define the 
hour-angle of the visible sun as that quantity of the celestial 
sphere's rotation which would carry the sun from the 
meridian to its actual position on the sky. And we may 
then define the apparent solar time at any instant as the 
hour-angle of the visible sun at that instant. Astronomers 
do not use a.m. and p.m. : apparent solar time is counted 
continuously from h to 24 h , like sidereal time. 1 

We have seen that successive returns of the sun to the 
meridian, giving the solar day, and successive returns of 
the vernal equinox, giving the sidereal day, are both caused 
by the same apparent axial rotation of the celestial sphere. 
We are therefore confronted by the question : why are these 
two kinds of day not exactly equal ? To answer this ques- 
tion, we recall (p. 27) that the sun appears at all times 
somewhere on the ecliptic circle in the sky ; but that (p. 29) 
it never appears at the same point of that circle on two 
successive days. 

The motion of our earth, in its annual orbit around the 
sun, makes us see the sun projected at opposite points of 
the ecliptic circle at intervals of about half a year. Opposite 
points of the ecliptic circle are 180° apart ; and half a year 
contains 183 days. Therefore, the sun changes its apparent 
position on the ecliptic circle about 180° in 183 days, or one 
degree daily. Now, to simplify matters, let us imagine 
that the sun appeared at the vernal equinox exactly at noon 
on a certain day. We already know that the sun appears 

1 Note 7, Appendix. 
68 



TIME 

at the vernal equinox once each year ; let us now imagine 
that it did so exactly at noon on one of the days in some 
particular year. On that occasion, the apparent solar day 
and the sidereal day must have commenced at exactly the 
same instant. For the one kind of day begins when the 
sun is on the meridian ; the other, when the vernal equinox is 
on the meridian. On the occasion when they were both on 
the meridian together, both days must have commenced 
together. 

But while the next apparent diurnal rotation of the sphere 
was in progress, the sun did not remain at the vernal equinox. 
Its daily change of about one degree, as seen projected on 
the ecliptic circle, must have made it appear approximately 
one degree east of the vernal equinox on the ecliptic, by the 
time a single diurnal rotation had been completed. There- 
fore, at the instant when the vernal equinox again reached 
the meridian, thus completing the sidereal day, the 
sun must still have been a short distance east of the 
meridian. The diurnal rotation must have continued a little 
longer to bring the sun to the meridian, so as to complete 
the apparent solar day as well. 

From these considerations it follows that the solar day 
is a little longer than the sidereal day. The difference is 
about four minutes : under the conditions imagined above, 
the sun would have reached the meridian at the end of the 
day about four minutes behind the vernal equinox. At the 
end of a second day it would have been about eight minutes 
behind the equinox, and so continuing on succeeding days. 

Thus there is a constantly increasing difference between 
solar and sidereal time, sidereal time gaining about four 
minutes daily on solar time. If a solar clock and a sidereal 
clock are placed side by side, it is easy to follow this con- 



ASTRONOMY 

tinually increasing gain of sidereal time by simply making a 
daily comparison between the two clocks. 

It is evident that this difference of the two clocks will 
amount to 24 hours in a year, since 4 m X 365 is approxi- 
mately 1440 minutes, or 24 hours. And the actual lag of 
the sun is a little less than 4 m , just enough to make the yearly 
gain exactly 24 hours. It is, in fact, evident that as the 
sun's apparent motion in the ecliptic circle is due to the 
earth's annual orbital motion around the sun, and as this 
orbital motion is completed in a year, it must happen at 
intervals of one year that the sun must return again to the 
vernal equinox, and everything repeat itself once more. 
The sidereal clock will gain just one day in the year; and if 
it agreed with the solar clock at the beginning of the year, 
the two clocks must again be together at the end of the 
year. Accordingly, the number of sidereal days in the year 
is one greater than the number of solar days. And the whole 
difference between sidereal and solar time is due to the fact 
that the sidereal day depends on the earth's axial rotation 
alone, while the solar day depends on both the axial rotation 
of the earth and the daily fraction of its annual orbital 
motion around the sun. 

This lagging of the sun behind the vernal equinox amounts 
to 4 m approximately each day, as we have seen, but this 
approximate quantity of 4 m is itself variable, within certain 
limits, throughout the year. The reasons for this variation 
will be explained in detail in a later chapter ; but one reason 
is quite obvious. The earth does not move uniformly in its 
annual orbit around the sun. And since the sun's apparent 
motion, as projected on the ecliptic circle, is simply a result 
of the earth's orbital motion, it follows that the sun's daily 
change of position in the ecliptic circle is not uniform either. 

70 



TIME 

Consequently, the lag of the sun behind the vernal equinox 
will not be the same each day, and as the sidereal days are all 
equal, because the earth rotates uniformly on its axis, the 
solar days are unequal. 

There are various inconveniences resulting from this in- 
equality of solar days : prominent among them is the diffi- 
culty of making solar clocks that will run with other than 
uniform motion. A clock keeping pace accurately with the 
inequalities of the solar day would be almost a mechanical 
impossibility. 

Therefore astronomers have adopted an imaginary con- 
ventional mean solar time, and a conventional unit for it, 
the mean solar day. These are so arranged that they corre- 
spond accurately to the average performances of the actual 
visible sun and the apparent solar day, and differ as little as 
possible from them. The mean solar days are all of equal 
length. We can, if we choose, even think of an imaginary 
mean sun in the sky, whose hour-angle from the meridian at 
any instant will be the mean solar time at that instant. Such 
a mean sun would occasionally have a greater hour-angle 
than the actual visible sun, and then the mean solar time 
would be later than the apparent solar time. The mean 
solar clock would be fast of an apparent solar clock, if 
there were such a thing. And when the mean sun's hour- 
angle was less than that of the visible sun, the mean solar 
clock would be slow. We shall return later to the difference 
between these two kinds of solar time more in detail : the 
above explanation is sufficient for our present purpose. 

These differences between mean solar time and apparent 
solar time are never greater than about one-quarter of an hour. 
But the difference between either kind of solar time and 
sidereal time of course ranges all the way from zero up to 

71 



ASTRONOMY 

24 hours. It is zero, as we have seen, when sun and vernal 
equinox are together. Then solar time lags behind sidereal 
time continuously about four minutes daily, until in a year 
the accumulation totals one day, and the two kinds of time 
are together again. We call the date in each year when the 
two kinds of time agree, March 21, or thereabouts. This is 
therefore the date where the sun appears in the vernal 
equinox. 

These facts explain clearly the varying aspect of the stellar 
heavens night after night. The fixed stars, as seen pro- 
jected on the sky, maintain positions practically unchanging 
with respect to the vernal equinox. Any fixed star will 
therefore rise, pass the meridian, and set a certain definite 
number of hours and minutes after the vernal equinox. 
In other words, it will do these things every night at the 
same sidereal time. Consequently, as the sidereal time 
gains about four minutes daily on solar time, each star will 
rise, pass the meridian, and set about four minutes earlier 
each night by solar time. 

For instance, referring to our table (p. 60), we find that at 
New York (approximate latitude 40°) Arcturus rises at 
9 p.m. on February 20. On February 21 it will therefore rise 
at 8.56; on February 22, at 8.52; etc. Two weeks after 
February 20, Arcturus will rise 56 minutes earlier, or approxi- 
mately one hour. This explains the statement (p. 59) that 
all the stars in the table will rise at 8 p.m. instead of 9 p.m. 
two weeks after the dates given in the table. 

Having now explained the meaning of time, it becomes 
possible to set forth very simply the astronomic signification 
of the time differences existing between different places on 
the earth. Why does Chicago time differ from New York time 
or London time ? Recurring to our definition of the celestial 

72 



TIME 

meridian (p. 36), we remember that it passes through the 
zenith, or point directly overhead. But the point overhead 
in London does not coincide with the point directly overhead 
in New York. Therefore London and New York will have 
different zeniths, and different celestial meridians. 

Furthermore, we have just explained solar and sidereal 
time to be the hour-angles of the sun and the vernal equinox 
from the celestial meridian. It follows that if London and 
New York have different celestial meridians, all hour-angles 
must be different at any instant in the two cities. Conse- 
quently, neither sidereal nor solar time at London will be the 
same as New York sidereal or solar time at the same moment. 
How much will they differ ? 

To answer this question we must have recourse once more 
to geography. The reader will remember that the surface 
of the earth is supposed to be divided by a series of lines called 
terrestrial meridians of longitude, great circles drawn on 
the earth from the north to the south terrestrial pole. We 
have already mentioned (p. 34), for instance, that the terres- 
trial meridian of Greenwich, England, is the prime meridian 
for reckoning terrestrial longitudes. And the longitude of 
New York is simply the angle at the north pole of the earth 
between the terrestrial meridians of Greenwich and New 
York. 

Now the celestial meridians of these two places cor- 
respond on the sky to their terrestrial meridians on the 
earth. 1 Therefore the angle between their celestial meridians 
at the north celestial pole will be the same as the angle 
between their terrestrial meridians at the north pole of the 
earth. In other words, it will be the same as their terres- 
trial difference of longitude. And since time at Greenwich 

1 Note 8, Appendix. 
73 



ASTRONOMY 

or New York is simply an hour-angle measured from the 
celestial meridian of Greenwich or New York, it follows 
that the difference in time will be equal to the longitude 
difference of these two places on the earth. 

Many beginners grasp this matter of time differences more 
easily in another way. Because the sun rises in the east, and 
moves westward in the sky, and because New York is west 
of Greenwich, the sun must pass the celestial meridian over 
Greenwich before it reaches that over New York. There- 
fore, when it is noon in New York, noon has already occurred 
in Greenwich, and it is already afternoon in the latter place. 
Consequently, Greenwich time is later than New York time ; 
and Greenwich clocks are fast of New York clocks. So of 
any two places, east clocks are always fast clocks : both 
words end in ast. 

To complete this part of our subject it is still necessary 
to explain what is meant by standard time, the ordinary 
time in actual use in our everyday affairs. It has no direct 
connection with astronomy, but is a mere conventional 
arrangement designed to prevent the inconvenience due 
to the fact that astronomical mean solar time, as we have 
seen, is practically never the same in any two places on the 
earth. It is not possible to avoid large time differences, 
such as exist, for instance, between Greenwich and New 
York. But there is no reason for the public to be troubled 
with minor time differences of a few minutes only. 

The plan actually adopted is as follows : Greenwich is 
taken as the initial point for reckoning all standard time. 
The earth is then divided by a series of standard meridians 
15° or l h apart, and everywhere the time of the nearest 
standard meridian is adopted arbitrarily for use instead of the 
mean solar time formerly employed. Thus our ordinary 

74 



TIME 

clocks not only fail to conform to the motions of the actual 
visible sun ; they no longer even run in conformity with the 
imaginary mean sun. But the standard time for which 
they are regulated differs from mean solar time by a constant 
difference only in each locality. This constant difference 
is the time difference already explained, as it exists between 
the terrestrial meridian of the locality and the nearest stand- 
ard time meridian. 

The great advantage of this system arises from the 
standard meridians having, by definition, time differences 
that are exact multiples of l h . The standard times of 
any two places must therefore differ by an exact number 
of hours, without minutes or seconds ; whereas the true 
mean solar time difference will practically always be an 
odd fraction of hours, minutes, etc. It follows, for instance, 
that a traveler going from New York to Chicago can set his 
watch on arrival by merely turning it back one hour. To 
make his watch accord with Chicago standard time, he 
does not need to consult any timepiece in Chicago. If 
his watch was correct in New York by New York standard 
time, it will be similarly correct in Chicago, if it be set one 
hour slow of New York time. 

We shall close the present chapter with a brief explana- 
tion of the International Date Line. This is another con- 
ventional arrangement intended to prevent certain difficulties 
arising from the time differences that confront travelers 
who circumnavigate the entire earth. A person going east- 
ward from Greenwich, for instance, will set his watch one 
hour faster for every 15° of longitude he traverses, in accord- 
ance with the explanations we have already considered. 

But if he should travel entirely around the earth, and 
continue the same treatment of his watch, he would find, 

75 



ASTRONOMY 

upon his return to Greenwich, that the watch had been set 
fast a total of 24 hours during the trip. The traveler would 
apparently have gained a day ; and if he kept a daily journal 
or diary, he would find the current date in his journal one day 
later than the date printed in the London morning papers 
issued on the day of his return to Greenwich. And in a 
similar way, if the traveler had proceeded westward from 
Greenwich, his diary would have been one day "slow" of 
the London papers on his return. 

Of course there is no real gain or loss of a day. If the 
traveler went around the earth with uniform velocity, and 
made the circuit in 24 days, for instance, he would have 
changed his longitude 15° daily, since 15° X 24 = 360°. 
This would make his daily time difference just one 
hour. Therefore, while he would appear to gain a day in 
24 days, yet each of these 24 days would be only 23 hours in 
length : his apparent gain of one day would be offset 
exactly by his loss of one hour on each of 24 consecutive 
days. 

The above inconsistency is not convenient, even though it 
is apparent merely, not real. Therefore it has been agreed 
that navigators shall change their date arbitrarily by one 
day when circumnavigating the earth ; and that they shall 
make this change when they reach a certain longitude, also 
arbitrarily chosen on the earth. The terrestrial meridian 
of longitude thus chosen is 180° distant from Greenwich. 
This meridian passes through the Pacific Ocean ; it is most 
appropriate for the purpose because comparatively few 
ships navigate that part of the earth, and so the arbitrary 
change need be made but rarely. 

But it has not been found possible to confine this change 
of date accurately to the 180° meridian of longitude. There 

76 



TIME 

are certain groups of islands crossed by this meridian, and it 
would obviously be most confusing to have different dates in 
force in neighboring islands of the same group. Therefore 
an arbitrary irregular line has been drawn on the map of 
the Pacific Ocean, and called the international date line. 
Navigators are all instructed to change their date by one day 
when crossing this line ; skipping a date if they are proceeding 
westward, and counting a date twice if they are moving 
eastward. And the arbitrary line is drawn in such a way as 
to avoid as far as possible confusing changes of date in 
neighboring islands or in the possessions of a single nation. 
It may be remarked also that some of the ordinary standard 
time meridians have been similarly bent a little at certain 
points, so as to avoid having two kinds of standard time 
in two parts of a single city, or in two cities very near each 
other. 



77 



CHAPTER V 

THE SUNDIAL 

By means of the definitions and explanations contained 
in Chapter IV, we can now solve a very interesting practical 
problem. The sundial is no longer an instrument of essential 
importance in everyday affairs, since time is now universally 
measured with mechanical clocks and watches ; but it still 
remains a most instructive toy, and is as much as ever a 
desirable ornamental monument in gardens and other public 
places. 

We shall confine our attention to one of the simplest 
forms of the instrument, — the dial drawn on a horizontal flat 
surface. Upon that surface is erected a vertical gnomon; 
and the shadow of this gnomon falling on the dial indicates 
the hour of the day by its position among the dial lines. 
Our problem is to design the correct shape of the gnomon 
and to draw the lines properly upon the dial itself. 

In Fig. 23 we give a sketch of a complete horizontal 
sundial. The gnomon abc is made of a piece of flat brass 
plate firmly fastened to the base ABCD, upon which the 
dial itself is drawn. The edge ab casts the shadow by means 
of which the dial measures time. 

It is necessary that the angle bac at the base of the gnomon 
be equal to the terrestrial latitude of the place in which the 
dial is to be used. And the gnomon may be designed easily 
so as to have the correct angle by the method shown in 
Fig. 24. 

78 



THE SUNDIAL 




Fig. 23. Horizontal Sundial. 

Draw the line ac of any desired length, according to the 
size of dial it is intended to con- 
struct. At the point c draw the 
line cb perpendicular to ac. The 
proper length of cb may be found 
by multiplying the length adopted 
for ac by the factor 1 given in the FlG - 24 - Rawing the Gnomon, 
following little table for various terrestrial latitudes : 

Table for Constructing the Gnomon 




Lat. 


Factor 


25° 


0.466 


30° 


0.577 


35° 


0.700 


40° 


0.839 


45° 


1.000 


50° 


1.192 


55° 


1.428 



Thus, in latitude 40°, if ac has been made 10 inches long, cb would be 
.39 inches. 

1 Note 9, Appendix. 
79 



ASTRONOMY 

This having been done, the gnomon will have the proper 
angle at its base. The construction of the dial itself is 
shown in Fig. 25. The double line ac corresponds to the 
line ac in Fig. 24, the two lines composing the double line ac 
in Fig. 25 being separated by the exact thickness of the 
brass plate used in making the gnomon. The gnomon 
must afterwards be fastened to the dial in such a way that 
ac of Fig. 24 will come exactly upon ac of Fig. 25. 

The hour lines of Fig. 25 are drawn as follows : continue 
the double line ac to a point M, and make the distance cM of 
such a length that it will be equal to the length of ac multi- 
plied by the factor 1 given in the following little table for 
various terrestrial latitudes : 

Table for Constructing Dial Lines 



Lat. 


Factok 


25° 


0.423 


30° 


0.500 


35° 


0.574 


40° 


0.643 


45° 


0.707 


50° 


0.766 


55° 


0.819 



Now draw the long line PcQ of indefinite length, perpen- 
dicular to ac ; and draw the two lines MN parallel to PQ. 
Draw the two circular arcs cc' with centers at M, and divide 
each arc into six equal parts, giving the points 1, 2, 3, 4, 5, 7, 
8, 9, 10, 11. Draw lines as shown : M 1, M 2, M 3, M 10, 
M 11, etc., and continue them to the line PQ, giving the 
points I, II, III, IV, 5', XI, X, IX, VIII, 7'. Then the lines 
a I, a II, a III, a IV, a\,a XI, a X, a IX, a VIII, a VII, as 
shown, will be the hour-lines of the dial for the several 
hours of the day. The six o'clock line is drawn from a to 
VI, parallel to PQ, 

1 Note 10, Appendix. 
80 



THE SUNDIAL 




81 



ASTRONOMY 

The hour-lines having been drawn in this way, and the 
gnomon fastened to the base as already indicated, the whole 
instrument is ready for use. When setting it up in the 
sunshine, however, it must be properly " oriented," or 
turned around to the correct position. This will be the case 
if the line ac is made to point in the exact north-and-south 
direction, the end c being toward the north. And the 
easiest way to orient the dial is to turn it until the shadow 
of the gnomon indicates the time in accord with a good watch 
previously set to correct time. 

But it must not be expected that the sundial will keep 
pace accurately with the watch. For the dial shows the 
shadow cast by the actual visible sun. And as the actual 
visible sun gives us apparent solar time (p. 67), the sundial 
must also give apparent solar time. 

The difference between this kind of time and mean solar 
time (p. 71) is shown in the following table for various dates 
in the year; and this difference should, of course, also be 
considered when orienting the dial by means of a watch. 

Table of Differences between Sundial Time and Mean Solar 

Time 



Jan. 1 


Dial slow 4 m . 


July 


1 


Dia 


I slow 3 m . 


Jan. 15 


ti 


slow 10 


July 


15 


it 


slow 5 


Feb. 1 


a 


slow 14 


Aug. 


1 


a 


slow 6 


Feb. 15 


n 


slow 15 


Aug. 


15 


a 


slow 4 


March 1 


a 


slow 13 


Sept. 


1 


n 


correct 


March 15 


a 


slow 9 


Sept. 


15 


a 


fast 5 


April 1 


a 


slow 4 


Oct. 


1 


a 


fast 10 


April 15 


a 


correct 


Oct. 


15 


a 


fast 14 


May 1 


it 


fast 3 


Nov. 


1 


a 


fast 16 


May 15 


a 


fast 4 


Nov. 


15 


it 


fast 15 


June 1 


a 


fast 3 


Dec. 


1 


tt 


fast 11 


June 15 


n 


correct 


Dec. 


15 


a 


fast 4 



We have already stated (p. 71) that the detailed explana- 
tions of these varying differences between the two kinds of 

82 



THE SUNDIAL 

solar time will be found in a later chapter ; for our present 
purpose it is sufficient to use the foregoing tabulation with- 
out further comment. 

But we must not expect sundial time to agree exactly with 
our watches, even after we have made allowance for the 
above table of time differences. For our watches indi- 
cate standard time (p. 74), whereas the foregoing table 
merely corrects sundial time to make it accord with mean 
solar time. To ascertain the additional correction re- 
quired to transform the mean solar time into standard or 
watch time, we must know the longitude difference of the 
place where the dial is located from the nearest standard 
meridian. 

For instance, New York, in longitude 74° west of Green- 
wich, is 1° east of the nearest standard meridian, which 
is in 75° west longitude from Greenwich. Therefore New 
York local mean solar time is later (or fast) of the nearest 
standard meridian (p. 74). The difference will be 4 m , 
since 1° must correspond to 4 m if 15° of longitude correspond 
to l h . It follows that the sundial, even after the correction 
from our table has been applied, will still always be 4 m fast 
of standard time as used in New York. This final difference 
of 4 m should again also be considered in orienting a dial by 
means of a watch. 

The foregoing directions for making a sundial have been 
put in such form that any one can use them, even if entirely 
ignorant of astronomic principles. But the knowledge we 
have gained in Chapter IV should enable us to understand 
the sundial much more thoroughly. In the first place, we 
recall (p. 40) that the altitude, or angular elevation, of the 
north celestial pole above the horizon is exactly equal to the 
terrestrial latitude of the observer. Now we have made the 

83 



ASTRONOMY 

surface of our dial level, and constructed the base angle 
of the gnomon such that the time-measuring edge ab is like- 
wise elevated by an angle equal to the latitude. And in 
orienting the dial we also turned it around until the gnomon 
pointed exactly north. 

In other words, the dial and its gnomon are so arranged 
that the edge ab of the gnomon points exactly at the north 
pole of the celestial sphere. The gnomon's edge is therefore 
parallel to the axis of the celestial sphere ; since, as usual, we 
may neglect the tiny radius of the earth in comparison with 
the infinite distance of the sphere. It follows that the 
diurnal rotation of the celestial sphere will seem to take place 
around the edge of the gnomon. 

So the sun each day will also seem to perform its diurnal 
rotation around the edge of the gnomon. Now we have seen 
(p. 68) that the apparent solar time at any instant, or the 
hour-angle of the visible sun at that instant, is simply the 
quantity of rotation made by the celestial sphere since the 
sun was on the meridian. The sundial merely measures 
this quantity of rotation ; and thus becomes a measurer of 
apparent solar time. When the visible sun is on the meridian, 
the shadow of the gnomon falls, as it should, on the north- 
and-south line of the dial, marked XII. When the quan- 
tity of diurnal rotation is 15°, or one hour, the shadow falls 
on the line marked I ; etc. 1 

The accompanying Plate 4 is a photograph 2 of the largest 
sundial ever built. It was erected about 1730 by Jai Singh II, 
Maharaja of Jaipur, and restored in 1902 by order of 
the Maharaja Sawai Madho Singh. The huge gnomon, 

1 Note 10, Appendix. 

2 From The Jaipur Observatory and its Builder; by Lieutenant A. ff. 
Garrett, R. E., and Pundit Chandradhar Guleri. Allahabad, 1902. 

84 



THE SUNDIAL 

containing stone stairs, is 90 feet high, and its base is 147 
feet long. The shadow falls on a great stone quadrant 
instead of a level surface ; and the radius of the quadrant is 
50 feet. The shadow moves on the quadrant at the rate of 
two and one-half inches per minute. 



85 



CHAPTER VI 

MOTHER EARTH 

There was once an old professor of astronomy who used to 
begin a lecture on " the earth " by telling his students that 
the old Greek astronomers always assigned to the earth 
the gender feminine, probably because she was constantly 
leading them astray in their scientific investigations. And 
it must be conceded that any one beginning to study the 
earth in its astronomic relations with the rest of the universe 
would find it almost impossible to avoid being misled by his 
early observations. In fact, the very first thing we must 
learn about the earth is to unlearn almost everything we 
ascertain by the actual use of our eyes. 

For instance, if an ignorant person — a person ignorant of 
astronomy — were asked to examine the earth and to 
describe it, he would say it is a flat plain, roughened with 
hills and valleys, but still in the main a great plain. But an 
astronomer would be compelled to ask him to unlearn this at 
once, because the earth is really a big round ball or globe. 

And further direct examination of the earth by this ignorant 
person would lead him to another fact which he would con- 
sider certain. He would say the earth is stiff and steady, 
and that it does not move. Another thing for him to un- 
learn as quickly as possible ; for here again is mother earth 
a deceiver, for she is really whirling around on an axis once 
a day, and also speeding along in her annual orbit around the 

86 



MOTHER EARTH 

sun at the rate of about eighteen miles per second. And 
she has a number of motions and wobbles in addition to 
these. 

Now such an imaginary person is by no means to be 
regarded as an impossibility. Probably a majority of those 
who have inhabited the earth since the beginning have been 
thus ignorant; possibly a majority of those now living are 
nearly as ignorant. Some of the greatest names of antiquity 
are linked with conceptions of the universe quite at variance 
with facts now known ; many of the ancient philosophers 
were quite without knowledge of the earth's true motions. 
Pythagoras, who lived in the sixth century before Christ, 
or some of his disciples, were perhaps the first to introduce 
the idea of terrestrial motion into science. Copernicus, in 
his great work De Revolutionibus (published 1543), quotes 
the Pythagorean philosophers in support of his new theories. 

But it is not our purpose to trace the development of 
modern accepted ideas as to the earth's motions through 
the vast literature of the last four or five centuries ; we 
shall confine our attention to an 
explanation of things as they are. 
In the first place, let us consider the 
rotundity of the earth. There are 
a number of convincing arguments 
to prove that the earth is curved, 
and not a flat plain such as it ap- 
pears to be. It has been circum- FlG 26 Cur ^ e of the Earth, 
navigated many times, for one thing. (From sacroboscu?' sphaem, Edition 
And an even stronger proof of the 

earth's curvature is furnished by the appearance of ships at 
sea. When we examine a vessel approaching us from a dis- 
tance (Fig. 26), we always see the masts and sails before the 

87 




ASTRONOMY 

hull becomes visible; and this quite irrespective of the direc- 
tion from which the ship is coming toward us. This proves 
that the earth's surface is curved — is convex — in all 
directions. It proves that the surface of the earth slopes 
downward, as it were, in every direction from the point 
where the observer stands. And once granting that the 
earth is convex, its approximate sphericity is proven beyond 
a doubt by the shape of the shadow it casts into space on 
the occasions when eclipses of the moon occur. A vast 
number of such eclipses have been observed; and always, 
without exception, the edge of the obscured part of the lunar 
surface is curved, and curved as only the shadow cast by a 
spherical earth could possibly be curved. 

Next, as to the earth's axial rotation : how do we know 
that it turns daily on an axis passing through the terrestrial 
poles ? Strong doubts existed on this point at least as late 
as the time of Galileo, early in the seventeenth century. 
Thus we may quote the following from p. 244 of Salusbury's 
quaint translation of Galileo's Dialogue on the Two Chief 
Systems of the World (published by Galileo in 1632 ; Salus- 
bury's translation published in 1661) : 

"Salviati: 'As in the next place, to the instance against 
the perpetual motion of the earth, taken from the impossi- 
bility of its moving long without wearinesse, in regard that 
living creatures themselves, which yet move naturally, and 
from an inborn principle, do grow weary, and have need of 
rest to relax and refresh their members — ' 

"Sagredus (interrupts) : 'Methinks I hear Kepler answer 
him to that, that there are some kind of animals which refresh 
themselves after wearinesse, by rolling on the earth ; and that 
therefore there is no need to fear that the terrestrial Globe 
should tire, nay it may be reasonably affirmed, that it 

88 



MOTHER EARTH 

enjoyeth a perpetual and most tranquil repose, keeping itself 
in an eternal rowling.'" 

To-day, as in the time of Copernicus or Galileo, the obvious 
astronomical arguments are not logically conclusive. There 
is nothing to determine whether the diurnal rotation of the 
heavens, sun, moon, and stars, is produced by the sky turning 
around the earth, or the earth itself turning in the opposite 
direction inside the sky. 

Fortunately we have now good experimental proof that 
the earth really turns on its axis once in twenty-four sidereal 
hours. But, strange to say, this experimental proof did not 
exist until 1851. In that year the physicist Foucault per- 
formed a most striking experiment in the Pantheon at Paris, 
whereby it became possible for the spectators to see the 
earth, as it were, actually turning under their feet. This 
Foucault experiment, as it has since been called, is not diffi- 
cult to perform ; it has been repeated by many astronomers 
and physicists since the original observation was made, and 
always with the same result, favorable to the hypothesis of 
terrestrial axial rotation. 

Foucault suspended a very long pendulum consisting of a 
heavy ball attached to a wire free to swing in any direction. 
The only object in using a pendulum of great weight and 
length is to diminish the disturbing effects of possible 
air-currents in the room, and of other undesirable causes 
which might make the oscillations of a smaller pendulum 
vary from their theoretically correct position. 

When such a perfectly free pendulum is set swinging very 
carefully, it will continue to vibrate back and forth, until it is 
finally brought to rest by the friction of the surrounding air, 
and the resistance to bending of the wire by which it is 
suspended. But the direction in space of the plane of vibra- 



ASTRONOMY 

tion (the direction in which the wire moves back and forth) 
will tend to remain constantly the same, because no forces 
are applied to the pendulum at right angles to the direction 
of its swing; and it would require the application of such 
forces to alter the direction in space of the plane of oscillation. 
This principle, that a free-swinging pendulum will tend to 
oscillate in an unvarying direction, is the fundamental 
principle of the Foucault experiment. 

Now let us suppose for a moment that the experiment could 
be performed at the north pole of the earth. Suppose we 
could there set the pendulum swinging in the direction of the 
star Arcturus, for instance, and that we marked on the 
floor, under the swinging ball, the direction in which the 
oscillations commenced. Then, if there were no axial ro- 
tation of the earth, the pendulum would continue to swing 
back and forth, exactly over the same mark until it stopped. 
And it would always swing in the direction of the star 
Arcturus. 

But if the earth is turning under the pendulum, it will carry 
the mark on the floor around with it. And the pendulum 
still constantly continuing to swing toward Arcturus, there 
must result a visible rotation of the mark on the floor with 
respect to the direction of the pendulum's swing. This 
motion of the mark will keep pace exactly with the terrestrial 
axial rotation ; and after the earth has made a complete 
rotation in twenty-four sidereal hours, the mark must once 
more come exactly in line with the direction of the pendu- 
lum's oscillation. 

In any latitude other than that of the north pole, the state 
of affairs is not quite so simple. But it is certain that in 
any latitude whatever, if the earth is perfectly immobile, and 
has no rotation of any kind, there can result no motion 

90 



MOTHER EARTH 

whatever of the mark on the floor with respect to the pendu- 
lum. Once started over the mark, the pendulum must 
continue to oscillate over it. Yet whenever and wherever 
this experiment has been tried, large motions of the mark 
have been observed. Moreover, and most important of all, 
the rate at which the marks have been observed to move has 
always been found to agree accurately with the rate calcu- 
lated by theory 1 on the supposition that the earth rotates 
on its axis once in twenty-four hours. The conclusion is 
irresistible that our earth is really subject to such a rotation. 

We are not limited to the Foucault pendulum for an ex- 
perimental demonstration of terrestrial axial rotation. It 
was pointed out by Newton that we can test this question 
by the simple experiment of dropping a heavy object from 
the top of a tall tower, and noting exactly where it falls upon 
the earth beneath. Newton had received a letter (Decem- 
ber, 1679) from Hooke, asking for some " philosophical com- 
munication." In his reply he suggests the above experiment 
and says the falling body " will not descend in the perpen- 
dicular, but, outrunning the parts of the earth, will shoot 
forward to the east side of the perpendicular." 

It is obvious that if terrestrial rotation really exists, the 
top of the tower will move faster than the bottom because 
it is farther from the center of the earth, and so moves on a 
longer radius. Therefore, a body dropped from the top 
retains an extra eastward impetus in descending, and must 
strike the earth a little to the east of the spot directly 
under the point from which it was allowed to fall. It would 
not fall parallel to the string of a plumb-bob. 

Modern experiments on this principle, performed in 1831, 
were on the whole inconclusive in their results because it 

1 Note 11, Appendix. 
91 



ASTRONOMY 

was found impossible to avoid the interfering effects of air 
currents, and because the metal balls that were allowed to 
fall could not be prevented from being deflected a little one 
way or the other as a consequence of friction with the air. 
The errors introduced by these disturbing causes were large 
enough to mask almost completely the eastward deflection 
predicted by Newton ; but this deflection undoubtedly 
exists to the extent required by theoretic calculations based 
on the accepted hypothesis of terrestrial axial rotation. 

Having thus described the evidence which leads us to 
believe in the sphericity and diurnal rotation of the earth, let 
us next consider the methods by which its size and weight 
have been determined. Up to the present we have assumed 
as a first approximation that the earth is exactly spherical 
in form. Though this assumption is not quite accurate, we 
shall continue it a moment longer, and use it to explain a 
simple method of measuring the earth's size approximately. 

We have but to return to the process of Eratosthenes 
of Alexandria (250 B.C.), one of the ancients who believed 
the earth to be round. Eratosthenes used a method practi- 
cally equivalent to setting up a vertical post, and observing 
each day the length of its shadow cast upon a level surface. 
He was especially careful to measure the shadow when it was 
shortest each day. This occurs, of course, at noon, when 
the sun is on the meridian. Furthermore, the length of the 
short noon-shadow is not the same every day, for a very 
simple reason. We recall that the sun always appears at 
some point in the ecliptic circle (p. 27), and that during 
about half the year that point is located between the celestial 
equator and the north celestial pole (p. 43). During that 
half-year there must come a day when the sun appears 
in that point of the ecliptic which is farthest north from the 

92 



MOTHER EARTH 




celestial equator. This point is called the Summer Solstice ; 
the sun reaches it on or about June 21 of each year; on 
that date we have the longest day of summer ; the sun rises 
higher in the sky at noon than it does on any other date, 
and the noon-shadow of a post is the shortest of all the noon- 
shadows during the year. 

While the noon-shadow will thus be the shortest possible 
on June 21 everywhere in the northern hemisphere, it will not 
be equally short in all places. 
For, as shown in Fig. 27, the 
length of the shadow will depend 
on the angular distance of the 
sun from the zenith at noon. In 
the figure, Z is the zenith, S the 
sun on the meridian at noon, 
BC the post, and AB the length 
of the shadow. In a place where 
the sun is exactly overhead, in 
the zenith, the post will cast no 
shadow ; but with the sun at S, 
the shadow has the length AB. 
And the angular distance of the sun from the zenith, or the 
angle ZCS, can be found easily by measuring the noon- 
shadow length AB together with the height of the post BC, 
and then constructing a diagram like Fig. 27. 

Now Eratosthenes not only made observations of this 
kind at Alexandria, but he caused similar observations 
to be made simultaneously at another place called Syene. 
He was able to assure himself that the corresponding obser- 
vations were really made on the same day by using in both 
places the date when the short noon-shadow was the shortest 
of the whole year. 

93 



Horizontal 



A 



P/ane 



Fig. 27. 



A B 

Length of Short Noon- 
shadow. 



ASTRONOMY 

The line joining Syene and Alexandria was a north-and- 
south line, or terrestrial meridian of longitude, as we would 
call it to-day. Eratosthenes measured on the surface of the 
earth, with measuring rods, the linear distance between the 
two places, and found it to be, in Greek measure, 5000 
stadia. By combining this linear measurement with his 
shadow observations, he was able to ascertain the size of 
the earth, supposed to be spherical. Figure 28 shows how this 
was done. The circle represents the earth, 
with Alexandria and Syene situated at A 
and S. The zeniths of Alexandria and Syene 
lie in the directions of Z' and Z, respectively. 
The shadow observations showed that the 
sun, on the day when its shadow was shortest 
at noon, was exactly in the zenith at Syene, 
while on the same day at Alexandria the 
angular distance of the sun from the zenith 
was one-fiftieth part of a circumference, or 
7° 12/ as we should call it in modern angular 

Fig. 28. Eratos- 
thenes' Measure- measure. 

Earth. ° f the Now the sun ' s distance from the earth is 
so great that its rays falling on Alexandria 
and Syene may be regarded as parallel. Therefore these 
rays would come down to the point A in a direction 
parallel to ZS ; and so the angular distance 7° 12', measured 
at A, is equal to ACS, the angle at the earth's center 
between terrestrial radii drawn to Syene and Alexandria. 
In other words, Eratosthenes found that 5000 linear stadia, 
measured on the surface of the earth, correspond to one- 
fiftieth of the entire circumference. Consequently, the 
linear length of the earth's whole circumference must be 
50 X 5000 stadia, or 250,000 stadia. And from this 

94 




MOTHER EARTH 



Sourdon 



iMoreuil 



LMorttdidier 



Clermont. 



. Arbre de 
Boulogne 



ilonquiere 



Mareuil 



measurement of the circumference 
Eratosthenes could find the length 
of the earth's radius, also in stadia. 
Unfortunately, we do not know the 
length of his stadium in modern 
measures, and are therefore unable 
to judge the precision of his result. 

But this old method of Eratos- 
thenes is to-day still in principle 
the method used for measuring the 
earth ; though modified, of course, 
by modern instruments of precision, 
and modern methods of observing. 
Accurately stated, the process of 
measuring the earth, which is called 
Geodesy, consists of two separate 
and distinct operations. The first 
corresponds to the measurements 
Eratosthenes made on the surface 
of the earth between Syene and 
Alexandria for finding the linear 
distance between these two places. 

Two suitable fundamental points 
on the earth's surface are selected, 
and their relative positions, as well 
as the linear distance between them, 
are measured with the utmost pre- 
cision. This is accomplished by 
means of a survey called a geodetic 
triangulation. First, a chain of 
triangles is laid down on the earth, as shown in Fig. 29 ; and 
then, with very accurate surveying instruments, all their 

95 



iDammarrin 



'Mont lay 



Paris ! 



► Briez 



'Malvoisine 

I 

Fig. 29. Geodetic Triangula- 
tion. 
(From Picard's Degre du Miridien 
entre Paris et Amiens, Plate II, p. 116. 
Paris, 1740.) 



ASTRONOMY 

angles, and at least one side of one triangle, are measured 
with the greatest care. 

The triangles are usually laid down in such a way that the 
two fundamental points originally chosen are situated near 
the two ends of the chain of triangles, and preferably near 
the north and south ends. Then a north-and-south line is in- 
serted in the survey ; and thus the process of geodetic triangu- 
lation finally furnishes us with the precise linear distance 
by which one of the original points is north of the other; 
or, in other words, the modern equivalent of the 5000 stadia 
of Eratosthenes. This distance is now usually expressed 
in meters or in feet. 

In addition to the triangulation, the other operation, 
which corresponds to Eratosthenes' post and shadow obser- 
vations, is completed with precise astronomical instruments 
such as will be explained in a later chapter. For our present 
purpose, it is sufficient to remark that with the astronomical 
instruments in question it is possible to determine by observa- 
tion of the stars, and with very high precision, the exact 
terrestrial latitudes of our two fundamental end points. This 
having been done, the difference of the two latitudes, so de- 
termined, gives us, in degrees, an arc corresponding to the 
arc Eratosthenes measured with his shadows. 

Recurring to Fig. 28, we may now let the points A and S 
represent the two end points of the triangulation, supposed 
situated on a north-and-south line, or terrestrial meridian. 
The survey gives the linear distance AS ; and the astronomi- 
cal observation of the latitude difference gives the corre- 
sponding angle ACS at the earth's center. It is then easy to 
form the following proportion : Angle ACS : 360° : : linear dis- 
tance AS : linear length of entire circumference. 

By the aid of this proportion we can calculate the length 

96 



MOTHER EARTH 

(or number of feet) in the earth's circumference, and thence 
obtain the length of the terrestrial radius. 1 It is 3959 miles 
long. 

It is scarcely necessary to remark that operations of this 
kind for determining the size of the earth have been repeated 
frequently at many different parts of the earth's surface. 
Indeed, the importance of the problem warrants the expen- 
diture of almost endless time and trouble for its solution 
with the highest possible precision. 

And a most interesting result has been found from this 
frequent repetition of Eratosthenes' method. The radii 
obtained in different parts of the earth 
are not in exact accord. The earth may 
be considered spherical as a first approxi- 
mation, but as a first approximation 
only. 

When we measure, for instance, the 
number of feet in an arc corresponding FlG - ^^l^ oi 
to 1° of latitude difference near the 
equator of the earth, and again in a very high latitude near 
the north pole, we find the two numbers different. The 
polar degree is longer in feet than is the equatorial degree. 
This can be explained in one way only. The earth is not 
an exact sphere, but is flattened somewhat at the poles, so 
that the meridian section is shaped somewhat as shown in 
Fig. 30 (greatly exaggerated). 

It is obvious, of course, that the more flattened a circular 
arc is, the longer must be the radius of the circle. A little 
circle with a radius of one inch will exhibit considerable cur- 
vature even in a very short arc ; but a large circle, with a 

1 The radius of a circle can, of course, be computed easily from the cir- 
cumference by well-known mathematical methods. 
h 97 




ASTRONOMY 

radius of 100 yards, will show but very little curvature in a 
short piece of it. So the curvature of our earth at the poles 
is like that of a large circle ; near the equator it is like that 
of a smaller circle. 

Now this flattening of the earth at the poles is exactly 
what we should expect if the earth's form has been in- 
fluenced by its daily axial rotation ; and it is certain to 
have been so influenced, The rotation must produce a 
centrifugal force which would tend to make the particles 
of matter composing the earth move from the polar to the 
equatorial regions. The quantity of such motion, and the 
consequent quantity of flattening, must depend on the 
velocity of rotation. If the earth rotated several times as 
fast as it actually rotates, we should expect a considerably 
larger difference between the polar and equatorial diameters 
of the earth. 

Newton made an attempt to calculate the flattening of 
the earth by means of his newly discovered law of gravita- 
tion. But his result was not accurate; on account of cer- 
tain inherent difficulties of the problem, it can be solved 
best by actual observations rather than theoretical com- 
putations. In 1672, the astronomer Richer had already 
made a scientific expedition to Cayenne, and there found 
that his astronomical clock, which ran correctly at Paris, 
lost about two minutes daily. This was mainly due to the 
same centrifugal force by which the flattening of the earth 
is produced. Richer's clock was a pendulum clock. At 
Cayenne, near the equator, the centrifugal force must be 
near its maximum. For this force being due to the earth's 
motion of rotation, it will be greatest in places near the 
equator, which are whirling around rapidly in a large circle. 
Places near the pole are near the rotation axis, and have 



MOTHER EARTH 

therefore comparatively slow motion and moderate cen- 
trifugal force. 

So the centrifugal force at Cayenne, being large, and 
acting contrary to the gravitational force of the earth as a 
whole, diminished that pull upon Richer' s pendulum, and 
therefore made it oscillate slower, so that the clock "lost 
time." 

In spite of Richer's observation and Newton's calcula- 
tion, many scientific men doubted the polar flattening of 
the earth ; especially as certain French geodetic results did 
not accord with this theory. But in 1735-1744 Maupertuis 
measured a meridianal arc in Lapland, and Bonguer and La 
Condamine one in Peru; the comparison of these arcs left 
no doubt of Newton having been right. 

In comparatively recent years our knowledge of the 
earth's true shape has been extended greatly by entirely 
new methods which we have not yet described. The modern 
applications of Eratosthenes' plan have all involved trian- 
gulations extending in a north-and-south direction only. 
But it should be possible to employ with equal advantage 
similar geodetic surveys extending in an east-and-west 
direction. Only, in the latter case, the purely astronomic 
observations would involve a determination of the longi- 
tude difference between the two end stations of the survey, 
instead of their latitude difference. 

But astronomers had no means of measuring longitudes 
with a precision comparable to their measures of latitude 
until the introduction of the electric telegraph. If the two 
end stations are telegraphically connected, it is easy to 
send practically instantaneous signals from one station to 
the other. By means of these signals, accurate clocks, 
regulated by observations of the stars, and mounted at the 

99 



ASTRONOMY 

two stations, can be compared, and thus the time difference 
(p. 72) of the two stations determined within a small 
fraction of a second. And the time difference once known, 
the corresponding longitude difference is at once obtained, 
since 15° of longitude correspond to each hour of time 
difference. Furthermore, since it has become possible to 
determine the terrestrial radius by east-and-west triangu- 
lations, it follows that we can now use equally well trian- 
gulations extending in any direction whatever, provided we 
measure both the latitudes and the longitudes at the two 
end stations. 

Still another and quite different method of verifying the 
precision of results obtained from triangulations has been 
introduced in recent years. We have seen that the in- 
creased centrifugal force near the earth's equator, acting 
against the earth's gravitational attraction, tends to diminish 
the effect of the latter, and that a pendulum will therefore 
swing more slowly near the equator than it will near the 
poles. The quantity of this retardation can be calculated 
accurately from the known approximate dimensions of the 
earth, and its known velocity of axial rotation. 

But when such calculations are compared with actual 
observations of pendulums carried to different places on the 
earth, it is found that the retardation near the equator is 
larger than can be explained as a result of centrifugal force. 
The reason is obvious. On account of the earth's flattening 
at the pole, the pendulum is actually farther from the earth's 
center when carried to the equator than it is in high northern 
latitudes, near the pole. As gravitational attraction, accord- 
ing to Newton's theory, diminishes with any increase of 
distance from the attracting body, it follows that the earth's 
pull upon a pendulum will be a minimum at the equator. 

100 



MOTHER EARTH 

Consequently, we need merely carry a pendulum of un- 
varying length to high northern and to equatorial latitudes ; 
and compare with great accuracy its time of vibration. 
The difference, after correction for the effects of varying 
centrifugal force, will be a measure of the variations in the 
earth's gravitational attractive force, and will thus become 
a measure of existing variations in the length of the earth's 
radius. Very elaborate " pendulum surveys" of this kind 
have been made in recent years, and these verify the results 
of our latitude and longitude geodetic triangulations. 

We may therefore regard the earth's true shape as now 
known with considerable accuracy. But as this accuracy 
has increased, with the introduction of modern precision, 
minor irregularities in the earth's shape have been brought 
to light. The meridians of our planet are in the main 
ovals, such that, approximately : 

Equatorial diameter minus polar diameter _ 1 
Equatorial diameter 295' 

But these meridians are not ellipses of exact form. In 
recent years a new mathematical term has been introduced 
by geodesists to describe the true shape of the earth. They 
call the earth a Geoid ; and a geoid is defined as a surface 
everywhere perpendicular to the direction of the plumb-bob 
string, or the pull of gravity, and therefore everywhere 
coinciding with the mean surface of the ocean. The geoid 
surface coincides theoretically with the earth's surface; 
for it includes the effects of centrifugal force, as well as all 
possible variations of the direction in which terrestrial 
gravity acts, and of the pull which it exerts. 

Having thus indicated the methods employed by astrono- 
mers to measure Mother Earth, let us next consider the 
process of weighing her. And when we begin to speak of 

101 



ASTRONOMY 

weighing the earth, it becomes necessary to emphasize the 
distinction existing between the so-called mass of a body 
of any kind and its weight. Bodies have weight on the earth 
simply because of the gravitational pull of the earth upon 
them. And we have already seen that this gravitational 
pull is not everywhere the same, being greatest near the 
poles, where the flattening of the earth brings us nearest to 
the center. Consequently, we need some kind of a unit, 
analogous to a unit of weight, but one that is everywhere 
the same. 

The unit of mass is such a unit. The weight of a body 
is variable in different places, but its mass is everywhere 
the same. If we can determine its mass in one place, we 
know its mass everywhere. For instance, if we adopt as our 
unit of mass a certain standard pound that is preserved in 
the United States Government Bureau of Standards in Wash- 
ington, and if we wish to know the mass of a certain stone, 
we might carry it to Washington, and there weigh it in 
comparison with the standard pound. 

If it weighed exactly as much there as the standard pound, 
we should say it had a mass of one pound. Now anywhere 
else on the earth it would still weigh very nearly one pound, 
because the gravitational attraction exerted by the earth 
varies but little in different localities on its surface, the 
earth being so very nearly an exact sphere. But if that 
stone could be carried to the surface of the sun, where the 
solar gravitational attraction is about 28 times as great, 
on account of the sun's vast bulk, it would then weigh 28 
pounds; but its mass would still be only one pound, as 
before. Its mass having been found to be the same as that 
of the standard pound in Washington, it would be the same 
everywhere in the universe. 

102 



MOTHER EARTH 

So when we speak of weighing the earth, we mean, in 
precise language, determining its mass. So far as terrestrial 
man is concerned, there is no exactness in speaking of the 
earth's weight, since there is no such thing as weight, except 
in the case of bodies situated on the earth's surface and 
attracted by the earth. This state of affairs is by no means 
objectionable, because, for all practical purposes in as- 
tronomy, it is really the mass of the earth that we need to 
know. 

We need to know how strongly the earth exerts a gravita- 
tional pull upon the other planets in the solar system. And 
under Newton's law of gravitation this pull is proportional 
to the earth's mass. No such thing as weight enters into 
Newton's law anywhere. According to that law, two bodies 
whose masses are M and M' and whose distance asunder 
is D, — two such bodies attract each other with a force of 
attraction which may be indicated by the following simple 
formula : 

Force of attraction = _ o . 

D 2 

Stated in words, this formula means that between these 
two bodies exists an attractive force which is proportional 
to the product of their masses, and inversely proportional 
to the square of the distance between them. 

Various experimental methods have been used to measure 
the earth's mass, all depending on the following principle : 
we take some small object on the earth's surface, and com- 
pare the attractive force exerted by the earth upon that 
object with the corresponding attractive force exerted upon 
it by some other large terrestrial body of known mass. 
The attractive force exerted by the earth can, of course, be 
measured by weighing the chosen small object with an ordi- 

103 



ASTRONOMY 



nary balance ; that exerted by the large object of known 
mass must be ascertained by means of special experiments. 
But when we thus know the relative attractive forces exerted 

upon the same small object 
by the earth and the terres- 
trial body of known mass, 
we know the relative masses 
of the earth and that body, 
since attractive force is 
always proportional to the 
mass of the attracting body. 
Thus we arrive at a knowl- 
edge of the mass of the earth 
in terms of the large body of 
known mass. 

We shall first describe 
the so-called " Mountain 
Method/' used successfully 
by Maskelyne in 1774 in 
Scotland. 1 He selected for 
his terrestrial body of known 
mass a certain hill called Schehallien, 2 and made a very 
careful survey of the region surrounding it. Figure 31 

1 Maskelyne, "Account of Observations made on the Mountain Sche- 
hallien for finding its Attraction," Phil. Trans. Roy. Soc. LXV, Part II, 
p. 500. "Redde, July 6, 1775." 

Hutton, "Calculations from the Survey and Measures taken at Schehal- 
lien, etc.," Phil. Trans. Roy. Soc. LXVIII, Part II, p. 689. 

Maskelyne and Hutton carried out their calculations in such a way 
that the density or specific gravity of the earth was made the principal 
object of their researches. We have modified slightly their presentation 
of the subject, so as to make the earth's mass or weight the object sought. 
The two problems are identical, as we shall see further on. 

2 To find a hill suitable for his purpose, Maskelyne sent into Scotland 
a certain Charles Mason, who selected Schehallien after a long search. 

104 




Fig. 31. 



Mountain Method of 
Maskelyne. 



MOTHER EARTH 

shows his method of procedure. PQ is a portion of the 
earth's surface, here supposed spherical, and C is the center 
of the earth. SA and NB are two plumb-bobs hung on 
opposite sides of the mountain at two principal stations of 
the survey. The station N was chosen nearly due north 
of the station S. 

Owing to the gravitational attraction of the hill, both 
plumb-bobs were deflected toward it. Instead of pointing 
toward the center of the earth at C, they pointed toward 
C", a point situated between the center C and the surface PQ. 

Now it was possible to ascertain by observation both 
the angle C and the angle C. For the angle C is simply 
the latitude difference of the two stations N and S, since 
they are on the same terrestrial meridian, or north-and- 
south line. And this latitude difference would be one of 
the results furnished by the survey, which must make 
known the number of feet N was north of S. Then, know- 
ing the terrestrial radius, the number of feet corresponding to 
one degree of latitude was known, and so the exact number 
of seconds of arc in the latitude difference was also known. 

On the other hand, the angle C" was ascertained by means 
of astronomical observations at the two stations N and S. 
It was merely necessary to make the observations usual in 
astronomic determinations of terrestrial latitude. It is 
sufficient for our present purpose to mention here one 
peculiarity of observations of this kind. It is always neces- 
sary, in adjusting our instruments, to make use of a plumb- 
bob, or its equivalent, a spirit-level, to ascertain the direc- 
tion of the zenith (p. 36) directly overhead. 

This was the same Mason who was employed in 1763 by Lord Baltimore 
and Mr. Penn to survey the famous Mason and Dixon line to settle the 
boundary between Maryland and Pennsylvania in the American colonies. 

105 



ASTRONOMY 

Ordinarily, results obtained in this way are correct : 
but, in the present case, they were rendered incorrect by the 
presence of the neighboring hill Schehallien. The astro- 
nomical latitudes determined at N and S were necessarily 
both erroneous, and the errors were equal to the deflections 
of the plumb-bobs at the two stations. Then, when the 
latitude difference was derived from the astronomical obser- 
vations, it came out as the angle C, instead of the correct 
latitude difference C. In other words, the astronomic 
observations gave the latitude difference referred to the 
false zeniths indicated by the plumb-bobs deflected by the 
mountain, while the survey gave the correct latitude 
difference C. 

In the actual experiment, the difference between C and C 
came out 12" of arc, a quantity large enough to admit of 
easy measurement ; and thus the angular deflection of the 
plumb-bobs became known. It was next necessary to ascer- 
tain by measurement the mass or weight of the hill itself. 
This was accomplished by first computing its volume ap- 
proximately from the data furnished by the survey. Then 
borings were made into the hill, and specimens brought to 
the surface. These were tested to ascertain their " specific 
gravity," or weight per cubic foot, as compared with water. 
With this information at hand it was easy to find the mass 
of the hill. 

The distance of the hill from the plumb-bobs being also 
known, it now became possible to calculate how great must 
be the attractive force exerted by the hill on the plumb- 
bobs to produce the observed deflection of 12" appearing 
in the difference C — C. The attractive force exerted by the 
earth on the plumb-bobs was ascertained by weighing them 
in an ordinary balance ; and thus Maskelyne found the 

106 



MOTHER EARTH 

relative attractive forces of the earth and of the hill upon 
the same plumb-bobs. And the ratio of these two attrac- 
tive forces then made known the relative masses of the earth 
and of the hill. We have just seen that the mass of the hill 
was ascertained from the borings, etc. ; and so the mass of 
the earth finally became known, too. This great classic 
experiment gave the first knowledge as to the mass of our 
planet. 

Unfortunately, the result was not very accurate ; the diffi- 
culties inherent in the measurement and testing of the hill 
precluded the possibility of high preci- 
sion. Consequently, a few years later 
(1798), Cavendish 1 employed a method 
which can be entirely completed in a 
laboratory, and which, with various 
minor modifications, has since given us 
all the information we possess as to the 
earth's weight or mass. Indirectly, yet 



just as surely as if the earth could be IG ' 32 ' olslon aauce - 
placed in a gigantic scale-pan, is it possible to weigh the 
planet. 

The principal part of the Cavendish apparatus is called a 
Torsion Balance, shown in Fig. 32. A very light rod ab 
carries a small metal ball at each end. The rod is suspended 
at its middle point d by means of a very fine silk thread cd 
from a fixed support c. In recent instruments the silk 
thread is replaced by a fiber of quartz made by fusing the 
quartz and drawing it out to a microscopic fineness. 

The balance can be set in rotation about the supporting 
fiber cd, and will then oscillate backwards and forwards like 
an ordinary pendulum, until it is gradually brought to rest 

1 Phil. Trans. Roy. Soc, 1798, p. 469. 
107 



ASTRONOMY 

by the continued friction of the surrounding air. During 
these oscillations the rod, of course, remains horizontal, being 
exactly balanced at its middle point d. 

Before explaining the use of such a balance in weighing 
the earth, it is necessary to show how the so-called " con- 
stant" of the balance itself may be determined. This 
constant may be called the " torsional constant" of the 
balance; it is a measure of the quantity of force which 
must be applied to the balance in order to make it turn 
about the support cd. This quantity of force will, of course, 
depend on the thickness and stiffness of the fiber suspension 
cd. For when the balance turns, the fiber is twisted, and 
therefore the torsional constant will be large if the fiber is 
of such a kind as to resist a twisting effort quite strongly. 

The letter T is used to designate the torsional constant 
of any given balance. Accurately stated, T is the quantity 
of force required to turn the balance through unit angle, 
the said force being applied to the balance at unit distance 
from the center d. In our modern system of units : 

Unit of length is the centimeter, 

Unit of angle is 57.3°, 

Unit of weight is the gram, 

Unit of time is the mean solar second. 

Now it is possible to determine the constant T for any 
given torsion balance by observing its time of vibration; 
and this having been done, we may apply the balance to 
our problem. For this purpose, it must be mounted in 
such a way that its oscillations can be observed while it is 
under the influence of the gravitational attraction exerted 
by a couple of heavy lead balls brought very close to the 
little balls which are on the ends of the torsion balance rod. 

1 Note 12, Appendix. 
108 



MOTHER EARTH 






Figure 33 shows the apparatus, the reader being here sup- 
posed to examine it from above, looking down upon it along 
the direction of the supporting fiber cd (Fig. 32). 

In Fig. 33 the line ab shows the position in which the 
small balls a and b would finally come to rest after oscillat- 
ing, if the balance were allowed to oscillate quite undis- 
turbed by the proximity of the big lead balls. But if these 
latter are placed in the position A' and B f , their gravita- 
tional force will attract the little balls a and b, so that the 
final position of rest will be a r b f in- 
stead of ab. And if the big lead balls 
are placed at A" and B", the final 
position of rest will be a"b". 

In addition to the torsion balance 
the apparatus for the Cavendish ex- 
periment must therefore include two 
big lead balls, together with suitable 
mechanical arrangements for trans- 
ferring them conveniently from the 
position A'B' to the position A"B". 
This having been provided, it is pos- 
sible to ascertain by observation the distances a'a" and 
b'b" ' ; and this, together with our knowledge of T, will tell 
us the quantity of gravitational attractive force exerted by 
the big lead balls upon the little balls a and b. 

But the corresponding attractive force exerted by the 
earth upon these little balls a and b may be ascertained 
easily by weighing them in an ordinary balance, since weight 
is merely a result of the earth's gravitational attraction. 
Thus we return to the principle used by Maskelyne, and 
which we have already explained to be fundamental in all 
experiments of this kind. Having ascertained separately 

109 




Fig. 33. Cavendish Experi- 
ment. 



ASTRONOMY 

the attractive force exerted on the little balls by the big 
ones and by the earth, we have once more the ratio between 
the masses of the big balls and the earth, since these attrac- 
tions are proportional to the masses according to Newton's 
law. And since masses are in a sense only another name 
for weights, we have the ratio of the earth's weight to that 
of the big lead balls. 1 

The best result obtained in this way for the mass of the 
earth, from the average of several modern repetitions of 
Cavendish's experiment, is : 

6 X 10 27 grams. 

The size, shape, and mass of the earth having been deter- 
mined, it is easy to calculate its average density or specific 
gravity. This is, of course, simply the average weight of a 
cubic centimeter of terrestrial material as compared with a 
cubic centimeter of water. We have merely to calculate 
the earth's volume from its radius, which is extremely simple 
if we regard the earth as a sphere, and not very difficult, 
even if we take account of the polar flattening. 

Knowing the earth's volume, we can then compute the 
weight of an equal volume of water, and the ratio of the 
weight of this volume of water to the weight of the earth 
will be the earth's average density. Thus we see that the 
problem of weighing the earth is really equivalent to the 
problem of determining the earth's density. (Cf. p. 104, 
footnote.) 

In this way the earth's density is found to be about 5.5, 
which means that a cubic foot of average terrestrial material 
weighs 5.5 times as much as a cubic foot of water. 

Having now discussed the methods of ascertaining the 

1 Note 13, Appendix. 
110 



MOTHER EARTH 

mass of our earth, and the average density of the materials 
composing it, we shall next consider for a moment the 
structure of the earth's core. Our knowledge is here neces- 
sarily based on theoretical considerations only, it being 
obviously impossible to penetrate the earth's interior for 
the purpose of making actual observations. The deepest 
existing mines and borings pierce but a very short part of 
the outer terrestrial crust, when we consider that the radius 
of the planet is about 4000 miles. 

But such as they are, these mines and borings indicate a 
decided increase of temperature as we go deeper into the 
earth. The fact that such temperature increases are always 
found shows that there must be a steady supply of heat 
from the interior ; if there were not, the outer shell contain- 
ing the borings would speedily acquire a uniform tempera- 
ture. And we have further conclusive evidence of great 
interior heat from the volcanoes. 

Many theorists have held in the past that there is a central 
molten nucleus in the earth ; we now believe that the 
hot nucleus is solid. It is doubtless quite hot enough to 
be fused under ordinary circumstances ; but at the enormous 
pressure existing inside the earth, it is probably impossible 
for any substance to melt, even at a very high temperature. 
The strongest argument for believing in a solid earth, as 
against a molten earth having a thin solid exterior shell, 
is derived from the phenomena of the tides. The tidal 
rise and fall of the oceans is caused by the gravitational 
attraction of the moon. If there were but a thin shell of 
solid earth, it would be forced to rise and fall also, for it 
could slide on the interior fluid mass. And if the shell rose 
and fell with the water, we would not have observable tides 
along the coast lines of the continents. The earth's in- 
ill 



ASTRONOMY 

terior is therefore probably solid, with a rigidity about 
equal to that of steel. 

Under this theory we must regard the fluid lava ejected 
by volcanoes as derived perhaps from minor " pockets," in 
some way protected from the usual pressure, and therefore 
containing molten matter. Or we may imagine that the 
pressure of the crust may be diminished materially at some 
point for a time, whereby the solid matter immediately under 
that point might suddenly fuse and give rise to an eruption. 

A very remarkable phenomenon having a certain bearing 
upon the above theories is the Variation of Latitude. This 
was first proved observationally by Klistner in 1888, when 
he found that the latitude of the Berlin observatory was 
subject to slight changes. In the following year an expedi- 
tion was sent to Honolulu, while careful observations were 
continued simultaneously in Germany. It was found that 
the latitude of Honolulu increased when the German lati- 
tudes decreased, and vice versa. 

Since terrestrial latitude is merely angular distance from 
the terrestrial equator, it follows from the above that the 
earth's equator must be swinging in some way. And as the 
equator is everywhere 90° distant from the terrestrial poles 
at all times, it follows that the earth's polar axis must also 
be in motion. 

Later elaborate observational researches have shown that 
such is really the case. The earth's pole is really in motion, 
though the motion is quite small. A circle with a radius of 
50 feet would include all the pole's wanderings so far observed. 
Mathematical investigations show that this phenomenon 
indicates a solid but not quite absolutely rigid earth, thus 
affording a further verification of the accepted theory as to 
the solidity of our planet. 

112 



MOTHER EARTH 

We now pass from the interior of the earth to the part 
which is above the surface, — the atmosphere. This is a 
mixture of various gases, principally nitrogen and oxygen, 
with small amounts of carbon dioxide, water vapor, and 
various rare gases in most minute quantities. The entire 
atmosphere is part of the earth, and moves with it in its 
diurnal rotation and annual orbital revolution. 

Perhaps the most important function of the atmosphere 
is the distribution of sunlight in all directions by reflection 
from the tiny particles in the atmosphere. This explains 
our being able to see objects on the earth by the help of 
sunlight. We cannot see such objects unless sunlight falls 
on them in the right direction to be reflected back from the 
object to the observer's eye. And as the atmospheric par- 
ticles reflect sunlight in all directions, it follows that some 
light is sure to fall on all surrounding objects in such a way 
as to be reflected to our eye and make the objects visible. 

This same cause produces the apparent bright back- 
ground of the sky in daytime. Were it not for the atmos- 
phere, the sky would be dark in the daytime, as it is at 
night ; and we should see the stars at all hours. And the 
blue color of the sky, as well as the other colors seen at 
sunset, etc., are doubtless a result of prismatic effects pro- 
duced by atmospheric particles. 

Twilight is another important phenomenon due to the 
atmosphere. After the sun has set below the horizon, it 
continues to illuminate particles of the upper atmosphere. 
These particles once more reflect the light, so that a certain 
diminishing quantity of atmospheric illumination continues 
until the sun has sunk about 18° below the horizon. 

Another function of the atmosphere is to act as a kind 
of blanket to retain solar heat upon the earth. The sun 
i 113 



ASTRONOMY 



sends us rays that are practically all light-rays. Rays of 
this kind pass quite easily through the atmosphere, and 
heat the earth's surface. Then, at night, when the earth 
begins to radiate heat into space, it sends out a kind of heat- 
rays that pass through the atmosphere with the greatest 
difficulty only. Consequently, the earth remains much 
warmer than it would otherwise do ; and this action of the 
atmosphere has much to do with making the earth habitable. 
The phenomenon is due to a transformation of the char- 
acter of solar rays by being first absorbed and then radiated 
by the terrestrial surface. The water vapor in the atmos- 
phere is particularly effec- 
tive in this matter. 

Another, less important, 
atmospheric effect is known 
as Refraction. Light-rays 
coming from any celestial 
body must pass through 
the air before they reach 
the observer. As shown in Fig. 34, these rays are bent, 
or refracted, as they pass from the outer, and less dense, 
parts of the atmosphere to the lower and denser strata. 
The light of a star in the zenith at Z would come straight 
down, without change. For it is a principle of refraction 
that in passing from any stratum to a denser one, light is 
not bent when it is perpendicular to the strata. But if it 
makes an angle with the surfaces of the strata, it is bent 
toward the perpendicular. 

Thus light coming from a star at S would travel through 
the air in a curve, and would finally reach the observer at 
as if it had come in a straight line from S'. The figure is, 
of course, greatly exaggerated ; but the effect of refraction 

114 




Surface of the Earth 
Fig. 34. Refraction. 



MOTHER EARTH 

is to make all the heavenly bodies appear to us nearer the 
zenith than they really are. The effect is greatest when 
we observe near the horizon. Thus the sun, when setting, 
will still be entirely visible after it has passed below the 
real horizon. At such a time, too, the lower edge of the 
sun, being nearest the horizon, is refracted more than the 
upper edge. And so the setting sun usually appears of an 
oval shape instead of round, as it should be. 



115 



CHAPTER VII 

THE EARTH IN RELATION TO THE SUN 

In the last chapter we have discussed the earth as a sepa- 
rate astronomic body, to be measured and weighed without 
special reference to any other object in the universe. We 
have also (Chapter II) considered the earth to some extent 
in its relation to the celestial sphere, and found how various 
important points and circles on that sphere correspond to 
the terrestrial poles, equator, etc. Finally, we have made 
use of the plane of our earth's annual orbit around the sun, 
extending it outward to the celestial sphere, to gain a defini- 
tion of the ecliptic circle (p. 27), and, for the purpose of a 
first approximation, we have taken the earth's orbit around 
the sun to be a circle, with the sun at its center (p. 25). 

But the real terrestrial orbit around the sun is a slightly 
flattened oval or ellipse, with the sun at a point situated 
near the center of the oval, and called the Focus of the 
ellipse. These facts were first discovered by Kepler, who 
used a method to be described in a later chapter ; if it were 
necessary to establish their correctness to-day, by means 
of observations, it would be possible to do so in a very 
simple way. 

The necessary observations would consist in ascertaining, 
on many different dates, the exact position of the point at 
which the sun appears projected on the celestial sphere. 
In other words, we should measure frequently, with suitable 
instruments to be described later, the sun's declination and 

116 






THE EARTH IN RELATION TO THE SUN 



right-ascension (p. 34) ; these, as the reader will remember, 
define the sun's apparent position on the celestial sphere, 
precisely as latitude and longitude define the position of 
any city on the earth's surface. 

Now if we locate on a celestial globe (p. 37) these succes- 
sive points occupied apparently by the sun on various 
dates, we shall find that they all lie on a single great circle 
of the celestial sphere, which, as we have already seen (p. 27), 
is called the ecliptic 

FcU ptic Cir cle 

circle. And the fact 
that the sun's ob- 
served positions on 
the celestial sphere 
thus all lie on a 
single great circle, 
constitutes an ob- 
servational proof 
that the earth's 
orbit around the sun 
is really contained 
in a single plane, 
or flat surface. 

Let us next, in 
Fig. 35, resume Fig. 
3 (p. 28), drawing it, however, in a slightly modified way, 
with the earth's orbit greatly enlarged. But in spite of this 
enlargement, the reader must remember that the earth, sun, 
and entire terrestrial orbit together represent a mere dot 
in comparison with the infinitely distant ecliptic circle on 
the celestial sphere. 

Now, in this Fig. 35, let the large circle represent the 
ecliptic circle on the celestial sphere, and let S' represent 

117 




Fig. 35. Orbit of Earth. 



ASTRONOMY 

various points at which the sun appears projected, when 
observed on different dates. The true position of the sun 
in space is always at S. Now draw straight lines from these 
observed points S' through S, and continue them to certain 
other points E. 

We know that the sun is projected on the ecliptic circle 
at the points S' because the earth, in its orbital motion, 
occupies successively the points E. If we take S as the 
true position constantly occupied by the sun, it follows that 
when the apparent positions of the sun on the ecliptic circle 
are at the points S' y the earth's positions E will all be some- 
where on the extended lines S'S. But as yet we do not 

know where the points E are 
situated on those extended 
lines S'S. We know they are 
somewhere on those lines, but 
to know the true shape of the 
ofEarth \ earth's orbit we must ascertain 

Fig. 3d. Sun s Angular Diameter. 

the relative distances of the 
various points E from S by a different kind of observation. 

This can be accomplished by measuring, with a suitable 
instrument, the Angular Diameter of the sun on the various 
dates when the positions S' were observed. To understand 
what is meant by angular diameter, let us imagine two 
straight lines, drawn from the earth to two opposite points 
on the sun's visible disk. Then the angle between those two 
lines is the sun's angular diameter. 

It is quite evident from this definition that the sun's 
angular diameter will be greater, in proportion as the sun 
is nearer to us. Figure 36 makes this quite clear. When 
the earth is near the sun, as shown at E 2 , the angular diameter 
is greater than when the earth is farther from the sun, as at 

118 




THE EARTH IN RELATION TO THE SUN 

E x . Consequently, if we have measured the sun's angular 
diameter corresponding to each terrestrial position E in 
Fig. 35, we can mark off the relative lengths of the dis- 
tances SE. Whenever the angular diameter was found to 
be large, we should make SE proportionately short, and 
vice versa. The first of the lines SE would be made of any 
convenient arbitrary length, according to the size chosen 
for the whole diagram. 

When all this has been done, the points E will represent 
various positions of the earth in its orbit. A smooth curve 
can be drawn through them, and it will be found to be, not 
a circle, but a slightly flattened oval or ellipse. The point 
S, occupied by the sun, will not appear at the center of the 
ellipse, but at the point already mentioned as being situated 
a little to one side of the center, and called the focus. 

But it is most important to notice that all this experi- 
mentation so far gives us only the true shape of the earth's 
annual orbit around the sun. It tells us nothing whatever 
about the actual size of the orbit in miles. This could not 
be otherwise, in the nature of things. For up to the present 
we have measured angles only; angular right-ascensions 
and declinations, and angular diameters. And it is a 
mathematical principle that angles alone can never make 
linear distances known. 1 

One more interesting fact might be verified experimentally 
by the methods we have just described. Referring again 
to Fig. 35, if the dates corresponding to the terrestrial 
positions E are taken into consideration, it will be found 
that the line joining the earth and the sun moves in a very 
peculiar manner. This line is called the Radius Vector. 
It is clear that it not only revolves around the sun as a sort 

1 Note 14, Appendix. 
119 



ASTRONOMY 

of pivotal point, but it also lengthens and shortens, accord- 
ing to the variations in the curvature of the terrestrial orbit. 

It will be found that the radius vector, in the course of 
these motions and changes of length, always sweeps over 
equal areas in equal intervals of time. If we take three 
positions of the earth E, such that the time-interval between 
the first and second is equal to that between the second and 
third, then the space or area included inside the orbit 
between the first radius vector and the second is equal to 
the corresponding space between the second and third. 
Each of these areas is a kind of triangle, of which two sides 
are radii vectores, and the third side is a bit of the curved 
orbit. These facts were discovered by Kepler in 1609, 
using, as we have said, a method of investigation quite dif- 
ferent from that here described. 

Having now attained a notion as to the shape of the 
terrestrial orbit, it is possible to explain one of the astronomic 
phenomena most important to man, — the Seasons. What 
is the cause of summer heat and winter cold ? 

For the moment we shall consider the northern hemisphere 
only. At a first glance, one might suppose that the curved 
shape of the earth's orbit would cause the seasons. For 
the sun not being accurately at the center, it must happen 
that we are nearer the sun when at some particular point 
of the orbit than we are at any other time. When at this 
point nearest the sun, called Perihelion, the earth, as a 
whole, does actually receive a maximum of heat. But this 
is masked so completely by another phenomenon that it is 
largely without effect in determining the seasons. In fact, 
the date of perihelion occurs about January 1 each year, 
so that we are actually nearest the sun in winter. 

The temperature at any given place on the earth depends, 

120 



THE EARTH IN RELATION TO THE SUN 

not on our slightly varying proximity to the sun, but on the 
relative duration of day and night. When we have long 
"days" and short " nights" ; when the sun is shining on us 
during more than half of each 24-hour day, — then is the time 
to expect hot summer weather. 

We have already learned in Chapter II that half the 
ecliptic circle on the celestial sphere lies between the celestial 
equator and the north celestial pole ; that the sun is seen 
in that northern half of the ecliptic circle during about half 
the year; and that during such half-year it is above the 
horizon daily for more than twelve hours. To be more 
precise, we found that at the times of the equinoxes, about 
March 21 and September 22, when the sun appears to 
cross the celestial equator, the days and nights are equal, 
and each is twelve hours long. But at the solstices (cf. 
p. 93), about June 21 and December 21, when the sun 
attains its greatest angular distance (or declination) north 
and south of the celestial equator, — at these solstices 
we have the longest and shortest days in the year, mid- 
summer day and midwinter day. 

But there is still another factor influencing this question 
of the seasons materially. As we have just seen, the earth's 
surface is heated more or less in proportion to the length 
of time the sun's rays fall upon it ; but it is also heated in 
proportion to the directness with which it receives those rays. 
In summer, the sun is not only above the horizon each day 
longer than in winter, but it is also higher up in the sky when 
it is above the horizon. Its rays therefore fall upon the 
earth more nearly vertically ; the sun not only acts during 
a larger number of hours, but it also acts more efficiently 
while the effect is being produced. 

The next important question in connection with the 

121 



ASTRONOMY 

seasons is to inquire as to the date when we may expect 
the hottest day of summer. We might at first think it 
should occur at the time of the summer solstice, about June 
21 ; and we do, in fact, on that date receive our maximum 
heat per hour and per day. But for a long time after that 
date the days continue longer than the nights; in each 
24-hour period the earth is heated more in the daytime than 
it is cooled at night ; it receives more heat than it radiates 
away into space, and is constantly becoming hotter. 

But as this process of increased heating continues, the 
earth, being hotter, acquires an increased capacity to give 
up or radiate heat in the night, because a hot body radiates 
faster than a cool body. At the same time, the daylight 
receipt of heat by the earth diminishes constantly as we 
leave the solstice date in June. So the daily accretion of 
heat is diminishing, because of the shortening of daylight ; 
the outgo is increasing, because of increased power of radia- 
tion ; and so there must come a time when a balance occurs, 
after which the earth begins to become cooler again. In 
the temperate regions of the northern hemisphere this 
happens about August 1, instead of September 22, the ap- 
proximate date of the autumnal equinox, which would be 
the date of balance if it were not for the hot earth's increasing 
capacity to radiate heat. After August 1 the night radia- 
tion begins to exceed the daily gain of heat, and the earth 
commences to cool, in anticipation of winter. 

In the southern hemisphere all these effects are reversed. 
There the south celestial pole is elevated above the horizon 
instead of the north celestial pole ; the southern half of 
the ecliptic circle corresponds to the long days and short 
nights, instead of the northern half ; and midsummer comes 
in December instead of June. 

122 



THE EARTH IN RELATION TO THE SUN 

And there is also another difference between the two 
hemispheres which is most interesting. We have already 
mentioned that the earth is nearest the sun about January 1, 
and that this causes a slight increase of heat, which we have 
so far neglected to take into consideration. In the southern 
hemisphere this little increase of heat occurs in summer, and 
so tends to make the southern summer somewhat hotter 
than the northern summer. 

On the other hand, the fact that the radius vector sweeps 
over equal areas in equal time-intervals indicates that 
the earth must move faster in its orbit when near the sun 
than when farthest from the sun. Another reference to 
Fig. 35 (p. 117) will make this clear : when the earth is near 
the sun, the triangles have short sides, and therefore the 
earth must move through a large angle in a given time-inter- 
val so that the short sides of the triangle may be compen- 
sated by an increase in the curved base, and the area thus 
maintained unchanged. It is a principle of mechanics that 
the orbital speed of any planet must be greatest when it is 
nearest the sun. 

The effect of this in the case of the earth is to make it 
traverse the perihelion half of its orbit seven days quicker 
than the other half. In other words, when the sun appears 
in the autumnal equinox point in September, we have to 
wait only about 179 days for it to reach the vernal equinox 
point in March. But the other half of the ecliptic circle, 
traversed apparently by the sun from March to September, 
requires about 186 days. These numbers may be verified 
by counting the days between these pairs of dates, taken 
from an almanac. 

It follows that summer in the southern hemisphere is about 
seven days shorter than summer in the northern hemisphere ; 

123 



ASTRONOMY 

and this just about balances the increased heat of the 
southern summer, which we have just seen is due to its 
occurring in the part of the year when the earth is nearest 
the sun. In the northern hemisphere, on the other hand, 
summer occurs when the earth is farthest from the sun ; but 
it occurs in the long half-year of 186 days. So there is 
an equalization of the summers in the two hemispheres. 
Both are about equally hot. The southern has slightly 
warmer days because of the sun's proximity, but it has 
seven less summer days ; the northern has slightly cooler 
summer days, but seven more of them. 

The case is different with the winters, as shown in the 
following schedule : 

Northern Hemisphere Southern Hemisphere 

Summer 186 days (far from sun) 179 days (near sun) 
Winter 179 days (near sun) 186 days (far from sun) 

From this it appears that the southern winter is seven 
days longer than the northern, and also that the southern 
winter days are of the cooler kind on account of increased 
distance from the sun. So there is no equalization of winter 
between the two hemispheres, as there is in summer. The 
southern hemisphere has a somewhat colder winter than the 
northern hemisphere; and the summers are approximately 
the same in both hemispheres. 

This interesting fact may be stated in a slightly different 
way : the difference between the average summer and winter 
temperatures must be greater in the southern than in the 
northern hemisphere. And this presents a much more 
important aspect of the whole question. If one hemisphere, 
taking the year as a whole, is somewhat colder than the other, 
can there not have been a remote age in the earth's past 
history when this difference was far greater than it now is ? — 

124 



THE EARTH IN RELATION TO THE SUN 

great enough, perhaps, to account for the vast glaciers of the 
geologic ice-age. 

Of course there is but one way in which the difference 
could ever have been materially greater than at present : 
there must have been a time when the terrestrial orbit 
was flattened in a greater degree than now, and when the 
sun was consequently much farther from the center of the 
orbit. But was there ever such a time, and, if so, what was 
the cause ? 

It is an obvious fact that the motions of our earth will 
not only be influenced by the gravitational attraction exist- 
ing between the earth and the sun, but also by that produced 
through the pull of the other planets. This latter effect 
is small compared with the solar effect ; but it is powerful 
enough to bring about certain very slow and somewhat 
irregular changes in the earth's orbit around the sun. 

But all these changes have one peculiarity : all are of the 
kind mathematicians call Periodic. That is to say, none 
can continue to act indefinitely in a single direction. Every 
part of the orbit that changes will change first one way and 
then the opposite way, so that after the lapse of sufficient 
ages of time, everything about the orbit must return again 
to its original form and condition. 

There is thus a peculiarly impressive perfection about 
the operation of Newton's law of gravitation in the solar 
system. No matter what changes are destined to occur, 
these changes will never disrupt the system mechanically. 
So far as gravitational forces alone are concerned, the solar 
system may endure forever. 

It may be of interest to give here the principal orbital 
changes of the above kind which have been brought to light 
by the labors of various mathematicians following Newton. 

125 



ASTRONOMY 

1. The orbital flattening undergoes slight changes with a 
period of 64,000 years. 

2. The angle between the celestial equator and the ecliptic 
circle (cf . Fig. 6, p. 35) varies slightly, with a period of about 
34,000 years. 

3. The longest axis, or diameter, of the oval terrestrial 
orbit is slowly twisting around the sky with a period of 
108,000 years. 

While we are considering these peculiar variations of 
long period produced by the complicated action and inter- 
action of gravitational forces, it will be of interest to describe 
briefly the famous phenomenon known as the Precession 
of the Equinoxes. To make this matter clear, it will per- 
haps be best to call attention to the methods probably used 
in ancient times to ascertain by observation the length 
of the year. In the first place, astronomers tried observa- 
tions by means of shadows. For instance, setting up a 
vertical pole, it is easy to fix the date when the shadow at 
noon is shorter than it is on any other date. This must, of 
course, occur on the day of the summer solstice (p. 93), 
when the sun appears highest in the sky. And the sun will 
then appear at that point of the ecliptic circle which is 
farthest north of the celestial equator. 

By counting the number of days until the same event 
occurs again, it is possible to obtain an approximate value 
for the length of the year. For the year is simply the 
period of time required by the sun to complete an entire 
circuit in its apparent motion around the ecliptic circle, 
due to the real circuit of the earth in its annual orbit around 
the sun. By counting in a similar way the number of days 
between two widely separated occurrences of the same obser- 
vation, it is easy to find the length of a considerable number 

126 



THE EARTH IN RELATION TO THE SUN 

of years joined together. In this way Hipparchus compared 
the date of the summer solstice fixed by Aristarchus of 
Samos in 280 B.C. with his own observation in 135 B.C., and 
thus found the number of days in 145 years. Dividing this 
by 145, he computed a very accurate value of the average 
length of the year. It was very nearly 365J days. 

Another method of ascertaining the length of the year 
was used by the Egyptians long before the time of Hippar- 
chus. They observed the phenomena called the Heliacal 
Risings of certain bright stars near the ecliptic circle. A 
star is said to have its heliacal rising where it rises above 
the horizon as near as may be at the time of sunrise. This 
can occur only on the date when the sun, in the course 
of its apparent motion around the ecliptic circle, happens 
to appear near the star in question. Star and sun will then 
rise together. By counting as before the number of days 
until the same event occurs again, it is possible to ascertain 
how many days the sun requires to complete an apparent 
circuit of the ecliptic circle from a given star back to the 
same star again. 

But the length of the year obtained in these two ways is 
not quite the same. The shadow year is a little shorter 
than that deduced from the method of heliacal risings. 

The sun, in its apparent motion, travels from a given point 
of the ecliptic back to that point again somewhat quicker 
than it proceeds from a given star back to the same star 
again. 

It is a very singular thing that the sun should thus move 
along the ecliptic faster than it moves among the stars. 
There is but one way in which this could possibly occur. 
The entire ecliptic circle, or at least the equinox and solstice 
points, must have some kind of motion among the stars. 

127 



ASTRONOMY 

In other words, while the sun is apparently traveling 
along the ecliptic circle, that circle must itself be moving 
slightly in the opposite direction, so as to accelerate the sun's 
apparent motion. Or, to be more exact, if the sun starts 
from the vernal equinox in its annual apparent motion, and 
moves exactly one degree along the ecliptic circle, it will 
then be a very little more than one degree distant from the 
vernal equinox. While the sun was moving its one degree, 
the equinox also moved a tiny distance in the opposite 
direction ; so that the distance from the vernal equinox to 
the sun is finally the sum of the two motions. 

Astronomers call the kind of year whose length may thus 
be determined by shadows the Tropical Year. It is the 
interval between two successive apparent returns of the sun 
to that point of the ecliptic circle which is farthest north of 
the celestial equator. When the sun reaches that point in 
its apparent course, it turns, and begins to move southward 
again. The point is a turning-point ; and the word "tropic" 
comes from a Greek word meaning "to turn." 

Hipparchus was able to measure also with considerable 
precision the length of the other year, — the period of time 
required by the sun to move in its apparent course along the 
ecliptic from a given star back to that star again. This 
kind of year, from its relation to the stars, is called the 
Sidereal Year. 

The difference between the two kinds of year is about 
twenty minutes, the tropical year being the shorter. Hip- 
parchus explained the difference correctly as a consequence 
of the annual motion of the vernal and autumnal equinox 
points. Now these points are merely the intersections of 
the ecliptic circle and the celestial equator on the celestial 
sphere. If they are in motion, such motion may be caused 

128 



THE EARTH IN RELATION TO THE SUN 

by a change in position of the ecliptic circle, or the celestial 
equator, or both. Hipparchus was able to show that the 
effect is produced by a slight motion of the celestial equator, 
the ecliptic remaining practically unchanged. The celestial 
equator is moving in such a way as to cause the equinoxes 
(its points of intersection with the ecliptic circle) to move 
along the ecliptic circle very slowly. 

Hipparchus had no difficulty in satisfying himself that 
the ecliptic circle did not itself change, and that only the 
equator and the equinox points were in motion. For 
his star observations showed that all the fixed stars main- 
tained constantly unchanged angular distances from the 
ecliptic circle on the sky; which could not have been the 
case if that circle was itself in motion. But the stars did 
change their angular distances (declinations) from the 
celestial equator. 

In fact, Hipparchus discovered these phenomena first 
from his star observations, which he compared with those 
of Timocharis and Aristyllus, made about 150 years before 
his day. From this comparison he ascertained the quantity 
of motion of the equinoxes, and thence computed the differ- 
ence in length between the tropical and sidereal years. The 
length of the tropical year he found, as we have seen, by 
means of shadow observations. The length of the sidereal 
year he then calculated by adding to the length of the tropi- 
cal year the difference between the two as he had com- 
puted it. This was, and is, the best method of procedure, 
as the length of the sidereal year cannot be observed directly 
with high precision. It was Hipparchus who named the 
motion of the equinoxes Precession. 

It is possible to explain the cause of precession by the aid 
of Newton's law of gravitation. We have already found 

k 129 



ASTRONOMY 



that the earth is not truly spherical, but that it is somewhat 
flattened at the poles. This amounts in effect to a spherical 
earth, with a girdle of protuberant material surrounding 
the equator. In other words, the earth has its biggest 
girth around the terrestrial equator. Figure 37 is intended 
to illustrate the existing state of affairs. It shows the 
spherical earth, with its north pole (N. P.), its equatorial pro- 
tuberance, and the 
planes of the equator 
and ecliptic. 

Now the sun and 
moon both exert a 
gravitational attrac- 
tion upon the earth, 
and also upon its 
equatorial protuber- 
ance. And, as we 
have already seen, 
the sun is always in 
the plane of the 
ecliptic ; we may add as a fact that the moon also happens 
to pursue an orbit that never goes very far from the same 
plane. But the lunar and solar attractions affect most 
strongly that part of the protuberant ring which is nearest 
to them. This tends to tip over the protuberant ring into 
the plane of the ecliptic. If no other forces were at work, 
the earth (Fig. 37) would simply revolve around an axis 
perpendicular to the paper and passing through the earth's 
center C, until the equatorial plane had been brought into 
coincidence with the ecliptic plane. 

The force which prevents this rotation is due to the 
diurnal turning of the earth on its axis. The earth is 

130 




Fig. 37. Precession. 



THE EARTH IN RELATION TO THE SUN 

trying to turn around two axes at once, — its rotation axis 
through the north and south poles, and the other axis we have 
just mentioned. The result is to produce what is called 
in the science of mechanics a composition of rotations. 
This leaves the earth turning around its regular rotation axis 
once daily, but makes that axis itself move in space in such 
a way that the celestial pole on its extended end revolves 
slowly on the sky in a circle around a fixed center called the 
Pole of the Ecliptic. This is a point on the celestial sphere 
90° distant from every point of the ecliptic circle. The 
celestial pole being merely the prolongation of the earth's 
rotation axis to the celestial sphere, and the rotation axis 
being set in motion by the composition of rotations, the 
celestial pole must evidently move on the sky. The pole 
of the ecliptic remains unmoved, because, as Hipparchus 
found, the ecliptic does not itself change. But the celestial 
pole, and consequently the celestial equator, are both subject 
to this precessional motion. 

The angular radius of the circle in which the celestial pole 
revolves on the sky around the ecliptic pole is equal to the 
angle between the celestial equator and the ecliptic circle, 
which is about 23^°. A complete revolution of the one 
pole around the other requires about 25,800 years ; for the 
annual precession of the equinoxes upon the ecliptic circle 
is 50.2" and a complete revolution of the pole must, of 
course, correspond to a complete revolution of the equinoxes. 
And we have : 

= 25,800, approximately. 

It must not be supposed that this precessional motion 
proceeds with perfect uniformity, for there are various causes 
of inequality. When the sun appears at the equinoctial 

131 



ASTRONOMY 

points in March and September, it is for the moment also 
in the celestial equator, because the two circles, ecliptic 
and equator, cross at the equinoctial points. At such times 
the sun does not tend to tip the earth's equator. But at 
the time of the solstices, when the sun is far from the equato- 
rial plane, it has its maximum tipping effect. The moon's 
effect is even more complicated, on account of certain 
periodic changes in the position of the moon's orbit. Thus 
the actual precessional circle marked out on the sky by the 
celestial pole really resembles a sort of wavy line, having 
about 1400 principal waves in an entire circuit of 25,800 
years. These waves are called the Nutation, or nodding, 
of the terrestrial axis. 

An interesting consequence of precession is its effect on 
the seasons in the northern and southern hemispheres. We 
have seen that the southern hemisphere is now on the whole 
colder than the northern, But after half a precessional 
cycle has elapsed, the northern will be the colder hemisphere. 
Thus the astronomical explanation of the geologic ice-age 
is made possible. For the ice-cap was in the northern 
hemisphere : it must have been formed at a time when 
precession made the northern hemisphere the colder one, 
and when, coincidently, the summer and winter halves of 
the year were unequal by much more than the present 
difference of seven days, on account of the periodic change 
of the earth's orbital flattening (p. 126). 

Another important result of precession is the fact that 
the celestial pole is not always near our present pole star. 
This star is now about 1J° distant from the true celestial 
pole ; in the time of Hipparchus it was 12° distant ; 12,000 
years hence Vega will be our nearest pole star; and 4000 
years ago « Draconis was the pole star. This is well shown 

132 







PLATE 5. Precessional Motion of the Pole, 



THE EARTH IN RELATION TO THE SUN 

in the accompanying Plate 5, reproduced from Hevelius' 
Prodromus Astronomies, Gedani (Dantzig), 1690. It contains 
the constellation Draco, as drawn by Hevelius, inclosed in 
the precessional circle, having the pole of the ecliptic at 
its center. The pole star appears at the end of the long 
tail of Ursa Minor, the Little Bear. The circle is divided 
into degrees, and it indicates that Hevelius observed the pole 
star at an angular distance of 4° from the celestial pole, 
which is situated at the lowest point of the circle. This is 
in very close agreement with the theory of the precessional 
motion of the pole, as explained above. 

Of peculiar interest, also, in this connection, are the 
theories held by Egyptologists as to the date of construction 
of the great pyramid. In that pyramid there is a long 
passage pointing due north, and elevated above the horizon- 
tal at exactly the right angle to view « Draconis when it was 
the pole star. There can be little doubt that this passage 
was purposely so built ; and there is therefore little doubt 
left as to the approximate age of the pyramid. 

There are still several other details in connection with 
the relation of earth and sun that we must consider here. 
For instance, we recall (p. 71) that astronomers use a 
mean solar day and a mean sun corresponding as accurately 
as possible to the actual performances of the real visible 
sun. It is now possible to make this relation between the 
mean sun and the real sun a little clearer. 

Since the length of the mean solar day represents the 
average of all the actual solar days, it is evident that the 
mean sun must be sometimes in advance of the actual sun, 
and sometimes behind it. The difference between the two 
suns cannot continue to increase indefinitely; as a matter 
of fact, the extreme value of the difference is sixteen minutes. 

133 



ASTRONOMY 

In other words, mean solar time may be as much as sixteen 
minutes fast or slow of apparent solar time. The difference 
between the two kinds of solar time is called the Equation of 
Time. The equation of time at any moment is defined, then, 
as the quantity of time we must add to the apparent solar time 
at that moment, to make it equal to the mean solar time. 

There are two principal causes producing the equation of 
time. The first has already been mentioned repeatedly. 
The earth does not move in its orbit with uniform velocity, 
but travels most rapidly near perihelion (pp. 70, 123). 
Consequently the sun, projected on the sky at its various 
apparent positions in the ecliptic circle, also appears to 
move other than uniformly in that circle. This, of course, 
puts the real sun in advance of the mean sun at times, and 
behind it at other times. 

But even if the real sun were projected upon the ecliptic 
circle with uniform motion, still there would not result an 
equality of the actual solar days. A reference to a celestial 
globe, or to Fig. 6, p. 35, shows that there is a variable angle 
on the celestial sphere between the ecliptic circle and the 
celestial equator. At the equinox points, where the two 
circles cross, there is an angle of 23 J° between them; but 
at the solstices, where the distance between the two circles 
is greatest, they are practically parallel for a short distance. 

Therefore even uniform motion in the ecliptic would not 
give uniform motion when projected on the equator. But it 
would require uniform apparent motion of the sun on the 
celestial equator to produce equality of the actual solar days. 
For we have seen (cf. p. 69) that the sun would have to 
move exactly the same distance on the equator each day to 
make all the apparent solar days exceed the unvarying side- 
real day by exactly the same amount. To repeat, then, 

134 



THE EARTH IN RELATION TO THE SUN 

the two causes of the equation of time are : first, variable 
motion of the earth in its orbit, producing variable apparent 
motion of the sun in the ecliptic circle; second, variable 
angle between the ecliptic circle and the celestial equator. 

We have already given (p. 82) a table of the equation 
of time for various dates in the year. It there appears as a 
table of errors of the sundial, because the dial keeps apparent 
solar time by the shadow of the actual visible sun, and must 
be corrected by the amount of the equation of time to make 
it conform to mean time. 

An interesting and frequently misunderstood consequence 
of the equation of time is the inequality of the mornings and 
afternoons at certain dates in the year. Morning begins 
at sunrise and ends at noon. Afternoon begins at noon 
and ends at sunset. Now sunrise and sunset occur 
when the actual visible sun appears or disappears at the 
horizon ; by convention, noon occurs when the mean sun is 
on the meridian. Thus the morning will be shortened and 
the afternoon lengthened by the amount of the equation of 
time, or vice versa. The difference on any date will be twice 
the equation of time on that date. 

In February the afternoons are about half-an-hour longer 
than the mornings; in November, they are half-an-hour 
shorter; on account of this effect of the equation of time. 
Furthermore, we have seen (p. 74), when considering stand- 
ard time, that the times in actual use in certain places 
may differ from their proper mean solar times by as much as 
half-an-hour. This again affects the difference between the 
morning and afternoon by twice its amount, or a full hour. 
On the dates mentioned above, it may happen that this 
hour is added to the half-hour arising from the equation 
of time; so that on certain dates and in certain places 

135 



ASTRONOMY 

morning and afternoon may differ as much as an hour and 
a half. It is easy to find an example in the ordinary almanacs. 
Thus for November 20, 1913, the almanac gives the standard 
times of sunrise and sunset at Detroit as 6.28 a.m. and 
4.7 p.m. This makes the forenoon 5 h 32 m long, while the after- 
noon lasts only 4 h 7 m . The difference is nearly an hour 
and a half . 

There now remains but one more phenomenon of impor- 
tance requiring attention in connection with our earth's 
annual motion around the sun. It is called the Aberration 
of Light, and was discovered by James Bradley, astronomer 
royal of England, in 1728. Bradley had been making some 
very precise observations of the declinations (p. 34) of cer- 
tain stars, and had found that observations made six months 
apart could not be brought into agreement. There was a 
slight displacement of the stars on the sky at the end of six 
months; after the lapse of a whole year they were back 
again in their old places. 

This matter puzzled Bradley greatly; for a long time 
he was quite unable to find any satisfactory explanation. 
Finally he came upon the solution of the problem while 
he was sailing in a small boat on the Thames. At the 
mast-head of the boat was a pennant ; and Bradley noticed 
that whenever the boat changed its course in tacking, the 
pennant changed its direction a little with respect to the 
river bank. There seemed no reason for this, because the 
wind was quite steady in direction. 

Then it occurred to him that the boat's own motion in- 
fluenced the pennant. Its position would be determined 
by a combination of the wind's velocity and direction, to- 
gether with the boat's speed and direction of motion. And 
he saw at once that light coming to us from a star would 

136 



THE EARTH IN RELATION TO THE SUN 

seem to come in a direction similarly depending on the 
true direction of the star, and the light's velocity combined 
with the direction and velocity of our terrestrial orbital 
motion. The earth is here the boat; and the aberration 
of light was explained. 

There is another familiar explanation which may make 
this phenomenon clearer. Imagine a person standing per- 
fectly still in a rain storm on a windless day. The drops 
will seem to fall perpendicularly downward ; but if the 
person runs rapidly, they will strike him in the face, precisely 
as if they were coming down in a slanting direction. The 
drops will seem to come towards the runner ; more exactly 
stated, the direction from which the drops seem to come 
will be thrown forward in the direction of the running ob- 
server's motion. 

In a similar way, the direction from which starlight seems 
to come is thrown toward that point on the sky toward 
which the terrestrial motion is for the moment aimed. 
We see the star a little too near that point. But the earth 
moves in a nearly circular orbit, and so is constantly chang- 
ing the direction of its motion. Therefore the aberrational 
change in the stars' positions is also constantly and similarly 
changing its direction. The final result is to make each 
star seem to describe a little closed curve on the sky, which 
is a sort of miniature copy of the terrestrial orbit around the 
sun. This little aberrational curve is, of course, different for 
different stars, depending on their positions in the sky 
with respect to the earth's orbit. And the reason why 
these aberrational curves are so small is that the velocity 
of light is very large compared with the earth's linear velocity 
in its orbit. For if light moved instantaneously, or if the 
earth had no motion, there would be no aberration. 

137 



CHAPTER VIII 

THE CALENDAR 

Perhaps the chief duty of astronomers has always been 
the orderly measurement of time ; not merely short intervals 
such as the hour and minute, but also the much longer periods 
represented by months and years. For the latter purpose 
various calendars have been devised. The most ancient 
were doubtless based on the motions of the moon, and 
were consequently very irregular and complicated. It will 
not be of interest to trace their development beyond the year 
45 b.c, when Julius Caesar put in force at Rome the form of 
calendar which bears his name, and which had been arranged 
for him by the Greek astronomer Sosigenes of Alexandria. 

The first thing to understand about a calendar in the 
modern sense is that every date, such as Wednesday, 
August 27, 1913, is composed of four different constituent 
parts : the day of the week, the day of the month, the name 
of the month, and the number of the year. We may then 
define the fundamental problem of the calendar thus : 
having given any three of these constituent parts of a date, 
to find the fourth. This is the problem we shall solve in the 
present chapter, both for the Julian calendar of Csesar, 
and the modern Gregorian calendar, now in general use. 
This calendar was named after Pope Gregory XIII, by whose 
orders it was introduced in 1582, though it did not receive 
recognition in England until 1752, and is not yet used in 
Russia. 

138 



THE CALENDAR 

Our fundamental problem may present itself in several 
different forms. For instance, an important event in Ameri- 
can history happened on March 4, 1865 ; on what day of the 
week did it occur ? This event was the second inauguration 
of Abraham Lincoln as President. 

This same event suggests a good illustration of another 
form in which our problem may present itself. Presidents 
of the United States are always inaugurated on March 4, 
at intervals of four years ; and, with rare exceptions, in years 
following a " leap-year." In what years during the twentieth 
century will these inauguration dates fall on Sunday ? 

A third form of the problem might be as follows : an old 
letter, of great historic interest, happens to have its date 
blurred so as to be partly illegible. Suppose we can read, 
however, that it was written in a certain year, and on the 
17th day of the month. It also appears from some re- 
mark in the letter itself that it was written on a Thursday. 
In what month in the year of the letter was the 17th a 
Thursday ? Such are the problems we can solve through a 
proper understanding of the calendar. 

The first difficulty that arises in devising a calendar 
comes from the odd lengths of the week and the year. We 
all know that there are seven days in the week, and we have 
learned that the year contains about 365£ days. And it is 
impossible to divide 365 { by 7 exactly, without a " re- 
mainder." Therefore the number of weeks in a year cannot 
be expressed as a whole number; this fact makes the year 
and the week " incommensurable/ ' as it is called. The 
difficulty could not be avoided by changing the number of 
days in the week, because no whole number of days, such 
as 6 or 9, can be an exact divisor of 365J. 

To bring about an exact division, it would be necessary 

139 



ASTRONOMY 

to change either the length of the day or the length of the 
year. But neither of these can possibly be altered, because 
both are natural units of time. The day (p. 66) is the 
quantity of time required by the earth to make one complete 
rotation on its axis. This quantity of time is fixed by nature, 
and is therefore called a natural unit. We have also arti- 
ficial or conventional time-units, such as the hour and minute. 
For instance, the hour is defined conventionally as one 
twenty-fourth part of the time required by the earth to 
complete one axial rotation. Being an artificial unit, it 
would be within our power to make the hour one twenty- 
fifth or one twenty-third part, and to have twenty-five or 
twenty-three hours in the day. This makes clear the dif- 
ference between an artificial and a natural unit of measure- 
ment : one is man's creation, and subject to change by him 
at will ; the other is fixed and unchangeable by nature. 

But chronology does not concern itself with minor sub- 
divisions of time, such as hours and minutes ; and the year 
of chronology, like the day, is a natural unit quite beyond 
our control. So we must perforce deal with the year and 
day as we find them; our artificial chronological units are 
the week and month. We have just seen that nothing would 
be gained by changing the number of days in the week ; 
we may add that it would be impossible practically to make 
such a change, even if it were desirable. Both the week 
and the month have acquired, from their antiquity, a species 
of historic changelessness which lends them a kind of per- 
manence almost as great as that possessed by the natural 
units themselves. 

We must next explain what is meant by the year in 
chronology. We have already had definitions (p. 128) 
of two different kinds of astronomic years. In chronology, 

140 



THE CALENDAR 

we use one only of these two time-periods, the tropical year. 
This is the interval of time between two dates when the 
sun, in its apparent motion around the ecliptic circle, attains 
its greatest angular distance from the celestial equator. 
When this occurs at the summer solstice (p. 93) we have 
the date when the sun climbs highest in the sky at 
noon, when shadows are shortest, when midsummer day 
occurs. 

These facts make plain at once the reason for using the 
tropical year in calendar making. Suppose we have become 
accustomed to midsummer day occurring on June 21. 
It is obvious that midsummer must necessarily happen 
when the noon shadows are shortest, etc. Now suppose 
(to exaggerate) that the calendar year differed by a day 
from the tropical year. If one midsummer day then fell 
on the calendar date June 21, the next midsummer day 
would fall on June 22. And each midsummer day would 
come a day later in its turn, until, after the lapse of a 
century or so, we should have midsummer in December, 
and our calendar would be completely reversed. The one 
absolutely essential thing is to have the calendar year as 
nearly equal to the tropical year as it is possible to 
make it. 

We have seen that the length of the tropical year can be 
determined easily by astronomical observations. It has 
been found to contain 365.2422 days; or, approximately, 
365J. Now the calendar year must of course contain a 
round number of days, without fractions ; the most obvious 
way to bring this about is to use a year of 365 days, and 
put in a leap-year of 366 days every fourth year. This is 
the Julian calendar already mentioned as having been put 
in force under Julius Caesar. 

141 



ASTRONOMY 
The error of this calendar is found easily as follows : 

Julian Calendar 

1st year 365 days 

2d year 365 days 

3d year 365 days 

4th year 366 days 

Total, 4 years, 1461 days 

Actual length of 4 tropical years (365.2422 X 4) 1460.9688 days 

Error of Julian calendar .0312 day in 4 years 

or .0078 day in 1 year 

The above simple calculation shows that the Julian 
calendar runs into error at the rate of 0.0078 day per 
annum. This amounts to one day in 128 years, and the 
Julian calendar will therefore pass out of accord with the 
true tropical motion of the sun at that rate. 

Another simple calculation shows how to correct this 
error almost exactly, and this leads to our present Gregorian 
calendar. It is clear that the Julian method of introducing 
a leap-year every four years somewhat over-corrects the 
error that would be caused by the use of a uniform year of 
365 days. We need to omit one of those leap-years every 
128 years. To do this most simply, it was decided under 
Pope Gregory to omit a leap-year once every century for 
three centuries; and in every fourth century to omit no 
leap-year. This omits three leap-years in 400 years, or 
one in 133 years, instead of one in 128 years, as required. 
And the Gregorian rule for leap-year then becomes the 
following : 

The year is a leap-year if the year number is divisible 
exactly by 4, without a remainder ; except that in the case 
of century years like 1500, 1600, etc., the divisor must be 
400 instead of 4. 

142 



THE CALENDAR 

Under this rule 1912 was a leap-year ; 1900 was not ; but 
2000 will be. 

Let us now calculate the error of the Gregorian calendar. 
In 400 years there are 100 leap-years in the Julian calendar. 
The exception in the Gregorian rule reduces this number 
of leap-years to 97. We therefore have the following cal- 
culation : 

Gregorian Calendar 

Number of days in 400 years (400 X 365) is 146,000 

and 97 leap-year days 97 

Total number of days in 400 calendar years 146,097 

Number of days in 400 tropical years (365.2422 X 400) 146,096.88 

Error of Gregorian calendar in 400 years .12 

This makes the Gregorian error in one year only .0003 
day; so that 3333 years will pass before there is an accu- 
mulated total error of a single day. This is an entirely 
negligible quantity, and so the Gregorian calendar may be 
regarded as perfectly satisfactory for all practical purposes. 
Having thus explained the construction of the calendar, 
the next step is to show how to calculate the week-day 
corresponding to any date in the past or future. Let us, 
for convenience, attach to the seven days of the week seven 
numbers, thus : 

Sunday, 1, 

Monday, 2, 

Tuesday, 3, 

Wednesday, 4, 

Thursday, 5, 

Friday, 6, 

Saturday, 7. 

Let us also designate as the " century number" the first 
two digits of the year number. Thus, in 1913, 19 is the 

143 



ASTRONOMY 

century number and 1913 is the year number. Then we 
have the following : 1 

Rule for Finding the Week-day 

1. Divide the century number by 4 and 7, and call the 
remainders resulting from the division the first and second 
remainders. 

2. Divide the year number by 4 and 7, and call the re- 
mainders the third and fourth remainders. 

3. Add five times the first remainder to the second re- 
mainder, and call the sum the " constant." 

4. Add the following five numbers ; viz. : the constant ; 
five times the third remainder; three times the fourth re- 
mainder ; the day of the month ; and the following number 
depending on the name of the month : 



in Jan. ; 


6, in 


ordinary years ; 5, in leap-years ; 


" Feb. ; 


2, in ordinary years; 1, in leap-years; 


" March 


; 2, in 


all 


years ; 


"April; 


5, " 


n 


a 


"May; 


0," 


(i 


11 -■'■'' 


"June; 


3, " 


a 


n 


"July; 


5, " 


tt 


tt 


"Aug.; 


1, " 


tt 


ti 


" Sept. ; 


4," 


it 


a 


"Oct.; 


6, " 


a 


a 


"Nov.; 


2," 


a 


a 


"Dec; 


4, " 


a 


tt 



And call the sum of the five numbers thus added the "sum." 
5. Divide this sum by 7, and call the remainder the fifth 

remainder. 

Then this fifth remainder, when increased by unity, will 

be the week-day number required. 

1 For a demonstration of this rule, see Note 15, Appendix. 

144 



THE CALENDAR 

As an example, let us find the week-day corresponding 
to July 4, 1913. We have : 

1. 19 divided by 4 gives first remainder, 3 ; 
19 divided by 7 gives second remainder, 5. 

2. 1913 divided by 4 gives third remainder, 1 ; 
1913 divided by 7 gives fourth remainder, 2. 

3. Five times first remainder, 15, 

second remainder, 5, 

The constant is 20. 

4. The constant, 20, 
Five times the third remainder, 5, 
Three times the fourth remainder, 6, 
Day of month, July 4, 4, 
The month number for Juty, 5, 
The sum is 40. 

5. 40 divided by 7 gives fifth remainder, 5. 

The fifth remainder increased by unity gives the week-day 
number as 6, corresponding to Friday. Therefore July 4, 
1913, is a Friday. 

The above rule applies to the Gregorian calendar ; but 
we may use it in the Julian calendar also if we simply omit 
the first and second remainders, and for the constant always 
use 0. 

The foregoing method of calculation may be replaced by 
a device called a Perpetual Calendar, by means of which 
all calendar problems may be solved with ease. The ac- 
companying form of perpetual calendar was arranged by 
Captain John Herschel ; it is convenient in use, and may be 
extended easily, indefinitely in either direction, backwards 
from 1860, or forwards from 1995, for which limiting dates 
it is here given. The months January and February appear 
twice in it ; the italicized January and February to be used 
in leap-years, which are also italicized in the columns of 
year numbers. In ordinary years the unitalicized January 

l 145 



ASTRONOMY 

and February are to be used. The calendar is Gregorian. 
The following examples will illustrate the use of this per- 
petual calendar in finding the fourth constituent part of a 
date for which three parts are given. 

1. What day of the week is July 4, 1913? Opposite 4, 
under Day of the Month, and in the column headed July, 
we find the letter F. We then find 1913 in the third ver- 
tical column of year numbers. Running up this column to 
the letter F, and thence turning to the right, we find Friday 
for the day of the week. This agrees with our former cal- 
culation by the rule. 

2. In what years following leap-years does inauguration 
day, March 4, fall on a Sunday? Opposite 4, and under 
March, we find the letter B. Opposite Sunday, B occurs 
in the first column. Consequently, March 4 is Sunday in 
1860, 1866, 1877, 1883, etc. But the only years in this 
column that follow leap-years are 1877, 1917, 1945, and 1973. 
In these years, therefore, inauguration day falls on Sunday. 

Having thus explained the civil calendar in ordinary use, 
we shall next, to complete the subject, describe the Ecclesi- 
astical Calendar as briefly as possible. The fundamental 
problem of this calendar is to find the date of Easter Sunday 
in any given year. Following Gauss, 1 we shall divest the 
subject of all non-essential details; and especially exclude 
the ancient terminology, which tends to involve this some- 
what complicated problem in unnecessary obscurity. 

Fundamental data are to be found in regulations adopted 
by the famous Ecclesiastical Council of Nice, which met in 
the year 325 a.d. According to decree of that council, 
Easter Sunday is the first Sunday that follows the first 

1 Gauss, Berechnung des Osterfestes ; v. Zach's Monatliche Correspondenz, 
August 1800. 

146 



THE CALENDAR 



Perpetual Calendar 



Dat or THE 
Month 


Jan. 
Oct. 


Apr. 
July 
Jan. 


Sept. 
Dec. 


June 


Feb. 
Mar. 

Nov. 


Aug. 
Feb. 


Mat 




1 8 15 22 29 


A 


B 


C> 


D 


E 


F 


G 


Monday 


2 9 16 23 30 


G 


A 


B 


C 


D 


E 


F 


Tuesday 


3 10 17 24 31 


F 


G 


A 


B 


C 


D 


E 


Wednesday 


4 11 18 25 


E 


F 


G 


A 


B 


C 


D 


Thursday 


5 12 19 26 


D 


E 


F 


G 


A 


B 


C 


Friday 


6 13 20 27 


C 


D 


E 


F 


G 


A 


B 


Saturday 


7 14 21 28 


B 


C 


D 


E 


F 


G 


A 


Sunday 




1860 
1866 

1877 
1883 
1888 
1894 
1900 
1906 

1917 
1923 
1928 
1934 

1945 
1951 
1956 
1962 

1973 
1979 

1984 
1990 


1861 
1867 

1872 
1878 

1889 
1895 
1901 
1907 
1912 
1918 

1929 
1935 

mo 

1946 

1957 
1963 
1968 
1974 

1985 
1991 


1862, 

1873 
1879 
1884 
1890 

1902 

1913 
1919 
1924 
1930 

1941 
1947 
1952 
1958 

1969 
1975 
1980 
1986 


1863 

1868 
1874 

1885 
1891 
1896 
1903 
1908 
1914 

1925 
1931 
1936 
1942 

1953 
1959 
1964 
1970 

1981 
1987 
1992 


1869 
1875 
1880 
1886 

1897 

1909 
1915 
1920 
1926 

1937 
1943 
1948 
1954 

1965 
1971 
1976 
1982 

1993 


1864 
1870 

1881 
1887 
1892 
1898 
1904 
1910 

1921 
1927 
1932 
1938 

1949 
1955 
1960 
1966 

1977 
1983 

1988 
1994 


1865 
1871 
1876 
1882 

1893 
1899 
1905 
1911 
1916 
1922 

1933 
1939 

1944 
1950 

1961 
1967 
1972 
1978 

1989 
1995 


What day 
of the week 
was 

March 4, 
1865? 

Under 
March, op- 
posite 4 is 
the letter 
B. In the 
1865 - col- 
umn, oppo- 
site B, is 
Saturday. 

Note. In 
leap-years, 
use Jan. 
and Feb. 

that are un- 
derlined. 



147 



ASTRONOMY 

full moon occurring on or after March 21, the date of the 
vernal equinox (p. 72). And it was further ordered that 
the day of the ecclesiastical full moon shall be the fourteenth 
day after new moon, or the day when the so-called ." moon's 
age" is 14 days. Our problem is to calculate the date of 
Easter Sunday in accordance with this regulation. 

Gauss' rule 1 is approximately as follows, some of the 
successive operations being identical with those already used 
for the ordinary calendar : 

1. Divide the century number by 4 and 7, and call the 
remainders resulting from the division the first and second 
remainders. 

2. Divide the year number by 4 and 7, and call the re- 
mainders the third and fourth remainders. 

3. Divide the year number by 19, and call the remainder 
the sixth remainder. 

4. Add five times the first remainder to the second re- 
mainder, and call the sum the " const ant." 

5. Add eight times the century number to the number 
13 ; divide the sum by 25, and call the remainder the eighth 
remainder. 

6. Subtract the first remainder from the century number, 
divide the difference by 4, and call the quotient (which will 
be a whole number) the result of operation 6. 

7. Add eight times the century number to the number 13, 
deduct from the sum the eighth remainder, divide what is 
left by 25, and call the quotient (which will be a whole 
number) the result of operation 7. 

8. Add the century number to the number 15, deduct 
from the sum the results of operations 6 and 7, and call 
what is left the result of operation 8. 

1 For a demonstration, see Note 16, Appendix. 
148 



THE CALENDAR 

9. Add the result of operation 8 to 19 times the sixth 
remainder, divide by 30, and call the remainder resulting 
from the division the seventh remainder. 

10. Add five times the third remainder, three times the 
fourth remainder, the constant, the number 2, and the 
seventh remainder; divide by 7, and call the resulting re- 
mainder the ninth remainder. 

11. Subtract the ninth remainder from the seventh re- 
mainder, and increase the difference by 28. The result will 
be the date of Easter Sunday in March. But if this date 
is greater than 31, Easter Sunday will fall in April, and its 
date in April will be found by subtracting the ninth re- 
mainder from the seven th> as before, and diminishing the 
difference by 3. 

The above rule applies to the Gregorian calendar. In 
the Julian calendar the same rule may be used; but the 
constant is then always 0, and the result of operation 8 
always 15. Furthermore, two exceptions to the above rule 
exist in the Gregorian calendar : 

1. When Easter Sunday comes on April 26 by the rule, 
April 19 must be substituted for April 26. 

2. Take eleven times the result of operation 8, and in- 
crease the product by the number 11. Divide the sum by 
30. If the remainder resulting from this division is less 
than 19, and if at the same time the seventh remainder was 
28 and the ninth remainder 0, the rule will give April 25 
for the date of Easter Sunday. When these conditions all 
occur, substitute April 18 for April 25. 



149 



ASTRONOMY 

As an example, let us calculate the date of Easter Sunday 
in 1913. We have : 

1. Century number 19 divided by 4, gives first remainder 3, 
Century number 19 divided by 7, gives second remainder 5. 

2. Year number 1913 divided by 4, gives third remainder 1, 
Year number 1913 divided by 7, gives fourth remainder 2. 

3. Year number 1913 divided by 19, gives sixth remainder 13. 

4. The constant is 20. 

5. The eighth remainder is 15. 

6. Result of operation 6 is 4. 

7. Result of operation 7 is 6. 

8. Result of operation 8 is 24. 

9. The seventh remainder is 1. 

10. The ninth remainder is 6. 

11. Easter Sunday is March (1-6 + 28), or March 23. 



150 



CHAPTER IX 

NAVIGATION 

The guiding of a ship across the unmarked trackless 
ocean is strictly a problem of astronomy : among the many 
problems of the science it is the most important commer- 
cially; certainly there is no other so astonishing to those 
who do not understand the simple methods employed for 
its solution. Briefly stated, the astronomic problem of 
navigation consists in ascertaining a ship's latitude and 
longitude by observing the heavenly bodies. If this can 
be done, we know the exact position of the ship on the 
earth's surface; and knowing also the latitude and longi- 
tude of the port to which the ship is bound, it becomes an 
easy matter to calculate, or to measure on a chart, whether 
the ship must be steered north, east, south, or west, in 
order to reach its destination. 

Inasmuch as ocean currents, leeway, or other causes may 
produce unperceived deflections as a ship moves through 
the water, it is necessary and customary for navigators to 
determine their position astronomically at frequent inter- 
vals; once each day, if possible. These successive astro- 
nomical observations furnish a continuous check upon the 
running of the ship. Each new observation gives a new 
" departure," as it is called; and helps to assure a correct 
" land-fall " at the end of the voyage, as sailors say. When 
clouds prevent satisfactory observations of the sky, the 
ship must be run by "dead reckoning" ; a very expressive 

151 



ASTRONOMY 

term which indicates how little confidence mariners have in 
their compass, as compared with observations of the un- 
erring heavens. 

Our concern is with the purely astronomic question of 
navigation ; with seamanship, knotting and splicing, charts 
and compass, leadline and sounding machine, we have 
nothing to do. And the astronomic problem is a very 
simple one, for both the latitude and longitude of the ship 
may be calculated if we measure with a suitable instrument, 
and in a suitable way, the angular elevation or altitude 
(p. 36) of the sun above the visible sea-horizon. 

The instrument used for this purpose at sea is called a 
Sextant ; it may be defined as a portable instrument for 
measuring the angular altitude of the sun above the horizon. 
Figure 38 will enable the reader to form an idea of its ap- 
pearance, and to understand its principle. The essential 
parts are two small silvered mirrors, M and m ; a telescope, 
EK; and a circle, AA, engraved with " graduations ' ; as 
they are called, by means of which angles may be measured 
upon it in degrees, minutes, and seconds. The mirror m 
and the telescope EK are firmly attached to the sextant ; 
but the mirror M is pivoted in such a way that it can be 
turned, and the angle through which it is turned measured 
on the circle by means of the index CB. When the mirror 
M is turned back until it is parallel to the mirror m, the 
circle reads Q , because the angle between the two mirrors 
is then 0°. In all other positions the circle measures the 
angle between the two mirrors. P and Q are sets of colored 
glasses, which can be interposed temporarily, when the 
sun's rays are so brilliant as to be hurtful to the observer's 
eye. R is a small magnifying glass, pivoted at S, intended 
to facilitate the examination of the index CB. At C and B 

152 



NAVIGATION 

are shown the "clamp" by means of which the index can 
be fastened to the circle, and the " tangent-screw," which 
will adjust it delicately, after it has been " clamped." / and 
F are accessories for the telescope. 

The mirror m has an important peculiarity. The silvering 
is scraped away at the back of the mirror from one-half its 
surface. Thus only one-half reflects; the other half is 




Fig. 38. The Sextant. 
(From Bowditch's Navigator, 1912 ed., p. 66, Bureau of Navigation, U. S. Navy.) 

simply transparent glass. A navigator looking into the 
telescope at E will therefore look through the mirror with 
half his telescope, and with the other half he will look into 
the mirror. 

Now it is a fact that half a telescope acts just like a whole 
one, always. If a person using an ordinary " spy-glass" 
covers half of the big end with his hand, he will see the same 
view he saw with the whole glass. Only, as half the light- 

153 



ASTRONOMY 

gathering power is cut off, this view will be fainter or dim- 
mer, — less luminous. Applying this fact to the sextant 
telescope, it is clear that the observer will see two things 
at once with the telescope : he will see what is visible 
through the mirror m with half the telescope; and with 
the other half he will see what is visible by reflection from 
the mirror m. 

If he holds the sextant in such a position that the telescope 
is horizontal, he will see the visible sea-horizon with half 
the telescope through the mirror. If the other mirror M 
is then turned to the proper position, and the sextant held 
in the hand with its telescope still horizontal, and its circle 
vertical, it is possible to see the sun at the same time with 
the other half of the telescope, the solar rays having been 
reflected from both mirrors. To make this possible, the 
horizontal telescope must, of course, be aimed at that point 
of the sea-horizon which is directly under the sun. The 
solar rays will then strike the mirror M first; be thence 
reflected to the silvered part of the mirror m; and finally 
into the telescope. So the observation consists in so adjust- 
ing or turning the mirror M , that the sun and the horizon 
can be seen coincidently in the telescope. 

The angle between the mirrors can then be measured on 
the circle ; and it is easy to prove * that the angular alti- 
tude of the sun will be twice the angle between the two 
mirrors. Thus the sextant becomes an instrument for 
measuring the sun's altitude; it remains to explain how a 
knowledge of that altitude will furnish us with the ship's 
latitude and longitude. 

To obtain the ship's latitude, it is best to measure the 
solar altitude when the sun is on the meridian, at apparent 

1 Note 17, Appendix. 
154 



NAVIGATION 

solar noon. Omitting certain very small corrections, the 
sun will then have its greatest altitude for the day ; so that 
the navigator need only begin measuring altitudes a few 
minutes before noon, and continue as long as the altitude 
is increasing. The moment it begins to diminish, he stops 
and "reads" his sextant circle ; thus obtaining the meridian 
altitude of the sun. The accompanying Fig. 39 shows how 
the latitude is obtained from such an observation. is 
the observer on the ship. The semicircle H'PSEH is that 
half of the celestial meridian (p. 36) which is above the hori- 
zon. H and H' are the south and north points of the hori- 
zon, where it is intersected 
by the celestial meridian. 
P is the north celestial 
pole (p. 31), and S the 
sun as observed on the 
celestial meridian. E, 90° 
from the pole, is a point H ° 

Fig. 39. Latitude from Observation. 

on the celestial equator, 

where it crosses the meridian. The angle SOH, or the arc 
£27, is then the observed altitude of the sun ; and the arc 
SE is the declination (p. 34) of the sun. This declination 
is always known ; it can be calculated in advance, because 
we know the annual orbit of the earth around the sun and 
the point of the ecliptic circle (p. 27) at which the sun 
appears on the date when the observation was made. In 
fact, the navigator always has at hand a copy of the 
Nautical Almanac, which is a book published annually by 
the United States government, in which the sun's declina- 
tion is printed for every day in the year. 

The navigator then simply subtracts the known declina- 
tion SE from the observed altitude SH, and thus obtains 

155 




ASTRONOMY 

the arc EH, or the altitude of the celestial equator above 
the south point of the horizon. As soon as EH becomes 
known, it is easy to obtain the latitude. For the arc PE 
is also known ; it is always 90°, because it is the angular 
distance from the equator to the pole. Therefore we need 
merely subtract PE and EH from 180°, to get PH', the 
angular altitude of the celestial pole above the horizon. 
But this (p. 40) is always equal to the latitude ; and so the 
latitude of the ship becomes known from the sextant measure- 
ment of altitude. 

In making observations of this kind, it is necessary to 
apply certain corrections to the observed altitude, of which 
the two most important are the correction for refraction, 
and the correction for semi-diameter. The former is due 
to the bending of the sun's light as it comes down to us 
through the terrestrial atmosphere (p. 114). The amount 
of this bending can be found in refraction tables which are 
printed in all books on navigation ; the navigator merely 
subtracts it from the altitude as actually observed. 

The other correction for semi-diameter is due to the 
fact that we cannot measure the altitude of the sun's center 
because the sun appears in the sextant telescope as a round 
disk, and it is impossible to estimate the position of its 
center accurately. Therefore navigators always measure 
from the lowest or highest point of the disk : in either case, 
the angular semi-diameter must be added to or subtracted 
from the observed altitude to get the altitude of the sun's ; 
center. This semi-diameter varies a little through the year 
because, as we know, the flattening of the earth's orbit 
around the sun (p. 118) alternately increases and diminishes 
the distance of the earth from the sun. But the exact 
value of the semi-diameter is printed for each day in the 

156 



NAVIGATION 

nautical almanac, whence the navigator obtains it, together 
with the sun's declination. 

To ascertain the ship's longitude, a somewhat different 
process is employed. In principle it depends upon the time- 
differences which, as we have seen, exist between different 
places on the earth (p. 72). Strictly speaking, longitudes 
are longitude differences. The longitude of New York, for 
instance, is really the longitude difference of New York 
from Greenwich. And the time difference between New 
York and Greenwich corresponds exactly to the longitude 
difference, one hour of time corresponding to each fifteen 
degrees of longitude. 

The navigator takes advantage of these facts in a very 
simple way. He carries in the ship one or more marine 
chronometers. These are merely very large watches, accu- 
rately made, and mounted in boxes with swinging supports, 
so as to prevent the ship's rolling from influencing the exact 
running of the instrument. Before leaving port, these 
ship's chronometers are " rated" accurately, by comparing 
them on successive days with standard telegraphic time 
signals from some astronomical observatory. 

By rating a chronometer we do not mean merely ascer- 
taining its error, or the number of minutes and seconds it 
may be fast or slow on a given date. Rating includes also 
the determination of the fraction of a second by which the 
chronometer increases or diminishes its error on each suc- 
ceeding day. For instance, if a chronometer is found to 
have the following error and rate : 

April 15, 1913, chronometer fast 28.0 seconds, and gaining 0.3 daily; 

then on April 30, 1913, fifteen days later, the chronometer 
would be fast 28.0 + 15 X 0.3 seconds, or 32.5 seconds. 

157 



ASTRONOMY 

Knowing the error and rate, the navigator can always ob- 
tain correct time from his chronometers, within the limits 
of accuracy with which they can be made to maintain a 
constant or unvarying rate. 

Marine chronometers are always set to Greenwich time; 
so that when a navigator takes the time from the chro- 
nometer, allowing for its rate, it is always Greenwich time. 
Now suitable sextant observations enable him to determine 
also the correct local mean solar time of the ship; this 
having been done, a simple comparison with the Greenwich 
time of the chronometer furnishes the time difference between 
the ship and Greenwich, and therefore also the longitude 
difference, or " longitude of the ship." 

It remains to explain how the ship's local time may be 
ascertained by observation with the sextant. This is ac- 
complished by observing the altitude of the sun, just as we 
have explained in the case of latitude determinations; 
only, while latitude observations are made at noon, time 
or longitude observations must be made rather early in the 
morning, or late in the afternoon. 

It is quite obvious that a measurement of the altitude or 
angular elevation of the sun in the sky must make the 
time of day known. For the altitude is zero at sunrise, 
and greatest at noon; consequently, if we know the alti- 
tude, we must be able to calculate 1 how far the sun has 
proceeded from sunrise toward noon in its apparent diurnal 
rotation across the sky. The calculation involves the use 
of spherical trigonometry, and cannot be explained in detail 
here; but enough has been said to show that such a cal- 
culation is possible. 

The methods given here for navigating a ship are the 

1 Note 18, Appendix. 
158 



NAVIGATION 

simplest and most easily understood. Many other methods, 
or modifications of the above methods, have been devised, 
and may be found in any standard book on navigation. 

Older methods are perhaps of minor interest, but the 
reader will surely wish to know how ships were navigated 
before the days of chronometers. The first chronometer 
capable of keeping reasonably accurate time at sea was not 
made until 1736, although it was in 1675 that Charles II 
issued his royal warrant establishing the office of astronomer 
royal, and making it the duty of that official to " apply 
himself with the most exact care and diligence to the rectify- 
ing of the tables of the motions of the heavens and the 
places of the fixed stars, in order to find out the so much 
desired longitude at sea, for the perfecting the art of naviga- 
tion." 

Without the chronometer the navigator could still obtain 
his local time, but he had no Greenwich time with which to 
compare it. But his latitude from a noon observation was 
always available, since comparatively rude instruments for 
measuring altitudes existed for centuries before the inven- 
tion of the sextant. Thus, in the early days, the navigator 
was forced to find his way with latitudes only. For in- 
stance, in a voyage from England to Rio, the ship would 
be steered southward and westward, more or less, until the 
" noon-sights' ' showed that the latitude of Rio had been 
reached. It was then merely necessary to steer due west, 
along the latitude parallel of Rio, and checking the latitude 
of the ship by daily noon-sights ; the lookout man forward 
would notify the navigator when he " raised the land." 
But with no knowledge of longitude, and in a sailing ship, 
the navigator might be uncertain by many weeks as to the 
date when he would reach the port of Rio. 

159 



CHAPTER X 

MOONSHINE 

The moon, more than any other celestial body, is in a 
very peculiar sense our own, for it is a satellite of the earthy 
revolving around us, and accompanying our annual orbital 
journey about the sun. Earth and moon together follow 
the same path, completing each year a full circuit around the 
sun. And the moon is important, too, in the history of 
astronomy : upon its peculiarly intricate motions ; its con- 
nection with eclipses; its lifting of the great waters of 
ocean in tidal ebb and flow, — upon a due explanation of all 
these things men have exercised their highest powers from 
the very beginning of the science. 

The moon is not self-luminous like the sun, but shines 
only by receiving light from the sun, and reflecting it to the 
earth (cf. p. 16). Its orbit around the earth, like most 
orbits, is a slightly flattened oval or ellipse, with the earth 
situated at the focus (cf. p. 116), a little to one side of the 
center. The orbital plane can be imagined extended out- 
ward indefinitely, so as to cut out a great circle on the 
celestial sphere, similar to the ecliptic circle (p. 27). Some- 
where in this great circle the moon will always be seen 
projected on the sky. The plane of the lunar orbit is in- 
clined to the ecliptic plane by a small angle, about 5° ; so that 
this is also the angle between the ecliptic circle on the celestial 
sphere and the great circle belonging to the lunar orbit. 1 

1 We learn in Spherical Geometry that the angle between any two 
great circles drawn upon a sphere is equal to the angle between the two 
planes in which the circles are situated. 

160 

o o 



MOONSHINE 

As the moon travels around the earth in its orbit, we see 
it projected on the sky, and apparently progressing around 
its orbital great circle, just as the sun appears to travel 
around the ecliptic circle. No wonder the ancients were 
puzzled when they saw both sun and moon alike moving 
around the sky in their respective great circles. Of course 
they thought both bodies were alike revolving around the 
earth, and concluded that the earth must be the immobile 
center of all things. But we now know that the moon 
appears to progress around the sky because it is really mov- 
ing around the earth, and we see this real motion projected 
on the sky. The sun, on the other hand, only seems to 
travel around the sky ; it is really stationary, and its motion 
is an apparent one, due to the real motion of the earth in its 
own annual orbit (p. 116). But to the eye both sun and 
moon alike seem to circle the sky. 

The angular velocity of lunar motion in the moon's pro- 
jected great circle is far greater than that of the sun in its 
ecliptic circle. Both bodies appear to move in the same 
direction, from west to east; but while the solar apparent 
revolution takes about a year, and therefore averages about 
1° daily, the moon completes a circuit from any fixed star 
back to the same star again in about 27} days, correspond- 
ing to an average daily angular motion of about 13°. This 
period of 27} days, from star to star, is called the lunar 
Sidereal Period, and corresponds to the sidereal year (p. 
128) of terrestrial orbital motion around the sun. But the 
moon has also another period, called the Synodic Period. 
To understand it, we must remember that as the moon's 
angular motion among the stars is about thirteen times as 
rapid as the sun's apparent angular motion, the moon must 
be constantly overtaking and passing the sun, much as the 

M 161 



ASTRONOMY 

minute hand of a watch is constantly overtaking the hour 
hand. The synodic period is defined as the interval of time 
between two such successive overtakings of the sun by the 
moon. It is about 29^ days long, or about 2\ days longer 
than the sidereal period. 

To explain this in a different way, suppose the moon and 
the sun are to-day both projected on the sky near a certain 
fixed star. Then, 27J days later, the moon will have circled 
the sky completely, and will be back near the same star. 
During the 27J days, however, the sun will have moved 
apparently some 27J° eastward, because of its apparent 
motion in the ecliptic circle. Therefore, to rejoin the 
sun, the moon will still need to travel those 27|° ; and 
this, at the rate of about 13° daily, will require approxi- 
mately 2J days. So the synodic period is again seen to be 
29£ days long. 

Probably the first astronomical phenomenon ever ob- 
served by man was the " waxing and waning" of the moon; 
its change in shape from a thin crescent, gradual, night 
after night, to the " half -moon" of Plate 3, p. 17; and 
finally its increase to the brilliant circular orb we call the 
full-moon. The accompanying Plate 6 is a photograph of 
the moon, nearly full; and the small additional picture is 
the crescent moon. The dim visibility of the remaining 
lunar surface within the crescent is explained on p. 164. 

What are these " phases" of the moon, and what is their 
cause? We have just seen that the moon is not self-lumi- 
nous, but shines by reflected sunlight. If the moon were 
incandescent, like the sun, we should see it always as a full- 
moon, or complete luminous circle. But it is a globe, and 
so only one-half its surface can be illuminated by the sun 
at any given moment. Now if the earth happens to be so 

162 




Photo by Barnard. 
PLATE 6. Full Moon and Crescent Moon. 



MOONSHINE 



placed that we can see the entire illuminated hemisphere, 
full-moon occurs. If the earth is so situated that we see 
only the unlighted hemisphere, the moon is wholly invisible, 
and we say it is " new-moon." 

Evidently, we shall see the illuminated hemisphere when 
we are on that side of the moon which faces the sun and 
receives light ; when we are on the side of the moon opposite 
the sun, we see the dark part. And as the moon goes com- 
pletely around the earth with respect to the sun in 29 J 
days, it must happen once in each such period that we are 
suitably placed for each of these phenomena. And, of 
course, at intermediate 
dates, we must be so 
placed as to see larger or 
smaller portions of the 
illuminated part, giving 
rise to the other visible 
phases. This is the 
simple explanation first 
found by Aristotle. 

It follows from the above, as shown in Fig. 40, that full- 
moon must always occur when the sun and moon are seen 
projected at nearly opposite parts of the celestial sphere. 
The figure shows how light from the sun illumines half of 
both earth and moon. To the inhabitants of the dark side 
of the earth, the sun is not visible, and it is night. But 
those same inhabitants evidently see the bright half of the 
moon, in the full-moon phase. Under these circumstances, 
the figure shows that the directions of the sun and full- 
moon, as seen from the earth, point toward opposite sides 
of the sky, approximately. 

It may be remarked, also, that if the sun, earth, and moon 

163 




Fig. 40. Full Moon. 



ASTRONOMY 

were always in a single plane, the earth, at the time of full- 
moon, would be exactly in line between the moon and sun. 
It would then cut off the solar light from the moon and give 
rise to the phenomenon called an Eclipse of the Moon. 
But we have already seen that the two orbit planes are 
not identical ; that there is an angle of 5° between them. 
It is this angle between the planes that prevents the occur- 
rence of an eclipse during every 29J-day period of lunar 
orbital motion, as will be more fully explained in a later 
chapter. Finally, Fig. 40 shows that the interval between 
two successive full-moons or two successive new-moons is 
the synodic period of 294 days, not the sidereal period of 
27 i days. For these phases must recur when the moon 
has made a complete revolution around the earth, measured 
by the sun, not by a star. 

Closely connected with lunar phases is the phenomenon 
called the " earth-shine," or "the old moon in the new-moon's 
arms," shown in Plate 6, p. 162, small photograph. It 
often happens that when the first slender lunar crescent is 
seen, a few days after the date of new-moon, the dark part 
of the moon, within the horns of the crescent, will be illu- 
minated faintly. This illumination of the dark part can- 
not come directly from the sun, under our accepted theory 
of lunar phases; nor can it be light from the moon itself, 
for we know the moon to be non-luminous. But it is ex- 
plained easily if we once more examine Fig. 40, p. 163. 
This figure makes clear that when we see the moon in the 
full-moon phase, the earth turns its dark side toward the 
moon. As seen from the moon, the earth is in the " new- 
earth" phase. 

All the earth phases, as seen from the moon, are opposite 
to the lunar phases, as seen from the earth. Thus, when 

164 



MOONSHINE 

we see the moon nearly new, as a slender crescent, the earth 
is nearly a full-earth to the moon. And the slight illumina- 
tion of the dark part of the moon, as we see it, is then due 
to the strong light thrown upon it by the brilliant full-earth, 
doubtless several times more luminous than the full-moon 
seems to us. 

The small photograph of Plate 6, p. 162, also gives a 
good opportunity to notice that the " horns' ' of the moon 
always appear to be turned directly away from the sun, as 
they are seen by us projected on the sky. This follows 
from the explanation of phases : we can understand it 
easily, if we paint a ball half black and half white, to repre- 
sent the moon, with half its surface illuminated by the sun. 
If we now hold this ball so as to see only a narrow sickle of 
the white half, we shall always find the horns of that sickle 
turned to the right, if the white half of the ball, which 
faces the sun, is turned to the left. 

Now the small photograph of Plate 6 was made by Bar- 
nard, at the Yerkes Observatory near Chicago, Feb. 14, 
1907, about one hour and twenty minutes after sunset, 
when the moon was very near the western horizon, where 
the sun had set. So, in Plate 6, if we imagine a line drawn 
between the two ends of the moon's horns, and a second 
line perpendicular to it, and passing downward in the Plate, 
this second line, if drawn far enough below the horizon, 
would pass through the sun on the sky. 

The small photograph, therefore, appears on Plate 6 just 
as it appeared to the eye when Barnard photographed it. 
The large photograph was taken with a different instrument 
at a different time, but it has been purposely turned around 
in Plate 6 to agree with the small photograph. This agree- 
ment may be verified readily by comparing the configuration 

165 



ASTRONOMY 

of markings on the two pictures. The photograph of Plate 3, 
p. 17, shows the moon as it would be seen on the meridian 
with an astronomical telescope; to make the large photo- 
graph of Plate 6 agree with it, it would be necessary to turn 
Plate 6 around through more than a right angle in the direc- 
tion in which the hands of a watch move. The configura- 
tion of markings would then again be in agreement. 

Having now explained briefly some of the lunar phenom- 
ena of phases and motions, let us next consider a pecul- 
iarity in which the moon differs absolutely from the earth. 
Astronomers have ascertained quite definitely that lunar 
air or atmosphere is altogether absent ; or, if present, exists 
only in an extremely attenuated form. The principal obser- 
vational proof of the non-existence of atmos- 
phere is derived from phenomena known as 
"occupations" of stars. We have seen that 
Fig. 4i. Oc- the moon, as it moves in its orbit around the 

cultations. 

earth, travels among the stars about 13° daily 
(p. 161). But the stars are very much more distant than 
the moon, though we see both stars and moon alike pro- 
jected on the background of the celestial sphere. 

Therefore it must happen occasionally that the moon 
passes between us and some individual star. In such a 
case that star is, of course, concealed from our view tem- 
porarily. Usually such "occupations" last about an hour, 
the duration varying according to the part of the moon the 
stars happen to meet. In Fig. 41, the moon moves across 
the sky in the direction of the arrow. The star S will there- 
fore be occulted longer than the star S', because it meets a 
wider part of the disk of the moon. 1 

1 The two lines shown in the figure, along which the two stars are about 
to be occulted, are called "chords" of the moon's disk. 

166 




MOONSHINE 

Now we find from telescopic observation that no matter 
where the occultation takes place, the disappearance of the 
star is always perfectly instantaneous; there is no gradual 
fading away ; it is blotted out with very striking suddenness 
while still in full brilliancy. If there were a lunar atmos- 
phere, we would surely see a progressive dimming of the 
star, particularly as it passed from the outer less dense 
layers of lunar air into the denser layers near the surface. 

Knowing, then, that the moon has no atmosphere, we 
must inquire what has become of it. For we now accept 
the plausible theory that the moon was once part of the 
earth, and that it was separated from the parent planet as 
a result of a continued and peculiar action of gravitational 
forces. In that case the moon must have taken some 
atmosphere with it when it left the earth. What has be- 
come of that atmosphere? 

The most plausible explanation of its loss is derived from 
the kinetic theory of gases. According to that theory, 
the molecules of a gas are in constant violent motion, and 
continually colliding with each other. If this was true on 
the outer confines of the moon's original gaseous atmosphere, 
it must have frequently happened that an outer molecule, 
after collision, bounced off in a direction away from the 
moon. It then encountered no other molecule, and was 
prevented from escaping into space by nothing but the 
moon's gravitational attraction. That is the only force to 
hold it. 

But the moon's gravitational attraction is comparatively 
slight, as compared with the earth's ; for, as we shall see 
later, the mass of the moon is only about ^ T part of the 
earth's mass; and gravitational attraction varies propor- 
tionally to the mass of the attracting body. Therefore it 

167 



ASTRONOMY 

is quite conceivable that the moon may have lost its at- 
mosphere by the kinetic method, while the earth, by reason 
of superior gravitational attraction, is able to retain it. 
However this may be, physicists are now agreed that there 
is ample molecular velocity to carry gases gradually away 
from the moon. 

Absence of atmosphere means also absence of water; 
for water, if present, would evaporate and form an atmos- 
phere. And without air and water, there can be no lunar 
inhabitants similar to ourselves. 

The shape of the lunar orbit around the earth, to which 
we have already referred, might be ascertained observa- 
tionally in the manner already explained for the earth's 
orbit around the sun (p. 117). It would merely be neces- 
sary to measure frequently the lunar angular diameter, and 
the moon's exact place as projected on the sky with refer- 
ence to the celestial equator; in other words, the moon's 
declination and right-ascension. This would enable us 
again to draw the outline of an orbit similar geometrically 
to the moon's actual orbit. But as in the case of the earth's 
path, observations of this kind give us no notion as to the 
actual size of the orbit in miles. To know this we must 
measure a linear distance somewhere, just as we found 
when describing the similar state of affairs in connection 
with the earth's path around the sun. 

We shall therefore next outline the method by which 
this may be done in the case of the moon. The easiest 
way is to observe the moon's position, as projected on the 
sky, simultaneously from two observatories widely sepa- 
rated on the earth. We can then use the known distance 
between the two observatories as a " base-line" for calculat- 
ing the moon's distance. Nor is it difficult to show that 

168 



MOONSHINE 

such calculations will make this distance known. It is 
found to be about 240,000 miles. 1 

It is interesting to note that we have now for the first 
time outlined a method of finding by observation the actual 
distance separating a heavenly body from the earth. We 
now see that astronomy can make measurements other 
than mere angular diameters and angular distances. Its 
grasp extends outward into space; by indirect methods, 
but methods perfectly valid, man has learned the distance 
of the moon just as though he could go there and measure it 
with a surveyor's tape-line. 

Closely related to the method of ascertaining the moon's 
distance is the mysterious word " parallax." The moon's 
parallax is defined as the 
angular semi-diameter or 
radius of the earth, as seen 
from the moon. Thus, in 
Fig. 42, AC is the earth's 
radius ; M is the moon ; and the small angle at M is the 
lunar parallax. 2 

The moon's distance (240,000 miles, in round numbers) 
is about 60 times the earth's radius. But of course the 
flattening of the lunar orbit makes the distance vary, just 
as we found was the case with the earth and sun, when we 
discussed the terrestrial seasons. Just as the earth has a 
perihelion, or nearest approach to the sun (p. 120), so the 
moon has a " perigee," or nearest approach to the earth. 
And the lunar orbit is more flattened than that of the 
earth; the actual distance of the moon may vary all the 
way from 222,000 to 253,000 miles. 

The axial rotation of the moon is a subject often found 

1 Note 19, Appendix. 2 Note 20, Appendix. 

169 




Fig. 42. Parallax of the Moon. 



ASTRONOMY 

puzzling, though really very simple. Here the crucial fact is 
derived from the most elementary telescopic observation 
of the moon. We find that the moon always turns approxi- 
mately the same hemisphere toward the earth. Whenever 
we look at the moon, we see the same configuration of surface 
details, lunar mountain ranges, etc. ; we never see the moun- 
tain ranges on the moon's opposite side. 

There can be but one reasonable explanation of this. 
The moon must have an axial rotation just rapid enough to 
produce this peculiar result. And here is the puzzle: 
many persons ask how the moon can have any axial rotation 
at all, if it constantly turns the same face toward us. The 
matter will be understood most easily by means of a simple 
experiment. Let the reader face a table in the middle of a 
room. Let him imagine himself to be the moon, and the 
table to be the earth. Let him now walk around the table 
in such a way that he faces it constantly. When he has 
gone halfway around the table, always facing it, he will find 
that he is looking at that wall of the room toward which his 
back was turned when he began the experiment. 

Thus he must have turned himself halfway around, while 
constantly facing the table. If his face was turned toward 
the north when he began, it is now turned toward the south. 
And if he completes a circuit of the table in the same way, 
returning finally to his original position, he will find that he 
has faced successively every point of the compass. This 
proves that he has turned himself around, or rotated once on 
his vertical axis ; yet, representing the moon, he has at all 
times turned his face toward the table, representing the 
earth. 

Accurately stated, the case stands thus : the moon makes 
an axial rotation in exactly the same time it takes to make an 

170 






MOONSHINE 



orbital revolution around the earth. We have seen that 
it revolves in its orbit in 27^ days ; it also finishes a rota- 
tion on its axis in 27J days. This is the whole explanation 
of the mystery. 

To complete this matter of the moon's rotation, we must 
now point out that the explanation, so far given, is not quite 
exact, though it is very nearly so, and quite sufficiently so 
for a first approximation. There is a phenomenon called 
Libration of the moon, which makes it possible for us to 
see a somewhat different part of the lunar surface at certain 
times. The lunar rotation axis is slightly inclined from 
perpendicularity to the plane in which is situated the orbit 
of the moon around the earth. The inclination is small, 
about 6 \° ; but it has the effect of tilting the moon, as it were, 
6^°, first one way and then the opposite way, according to its 
position in its orbit around the earth. For the lunar rota- 
tion axis remains constantly parallel to its original direction 
in space during the entire orbital revolution. On account 
of this tilting effect, we see a slightly different hemisphere 
of the moon at different dates in each lunar period. 

Still another libration of the moon exists. It is true that 
the moon rotates on its axis in the same period of time it 
requires for its orbital revolution around the earth; but 
while the axial rotation is uniform, the orbital motion, 
of course, is variable. As in all orbital motion, the velocity 
is greatest when the moon is in that part of its orbit which 
lies nearest the earth. Consequently, the axial rotation 
and the orbital revolution do not increase at precisely the 
same rate ; and from this cause, also, we see a slightly different 
hemisphere of the moon at different dates in the month. 

These are the two principal librations; there remain 
certain other very slight ones which we may here omit 

171 



ASTRONOMY 

as unimportant. But the combined effect of them all is 
as follows : 

1V0 of the lunar surface is always visible, 

T 4 oV of the lunar surface is never seen, 

T a o 8 o of the lunar surface is sometimes visible. 

It is of interest to note in passing that this agreement of 
the lunar axial rotation period with the period of orbital 
revolution is not due to chance. It must have resulted from 
some physical cause ; the theory at present accepted by 
astronomers considers it to be a result of forces, inter- 
acting between the moon and the earth, and analogous 
to those producing ocean tides. These forces probably 
brought about the existing state of things long ago, in 
early cosmic ages, when the moon may be considered to 
have not yet become perfectly solid, and to have therefore 
been subject to enormous tidal distortions. 

The next thing we have to do is to explain how astronomers 
measure and weigh the moon. Of course this cannot be 
done by the methods used in the case of the earth, because 
it is impossible to visit the moon to make surveys and 
perform the Cavendish experiment (p. 107) for determining 
mass. But the moon's size can be derived easily from 
measures of its angular diameter, combined with the knowl- 
edge we have already obtained as to its distance from the 
earth. 

In Fig. 43, the angle AEB is the moon's angular diameter 
(p. 118), as seen from the earth E. BE and AE are each 
equal to the known distance of the moon from the earth. The 
triangle ABE is therefore fully known, 1 and we can calculate 

1 All parts of a triangle can be calculated by trigonometric methods, if 
we know two sides and the angle between them. 

172 




MOONSHINE 

the number of miles in the lunar diameter A B. The 
average angular diameter is measured easily with astronomi- 
cal instruments; it is found to be about 31' of arc. This, 
combined with the known distance (about 240,000 miles), 
makes the moon's diameter about 2200 miles, or a little more 
than one-quarter of the earth's diameter. And as we know 
from geometry that the volumes of spheres are proportional 
to the cubes of their 
diameters, it follows 
that the volume of 
the earth is some- 

Fig. 43. Moon's Diameter. 

what more than 64 

times that of the moon (64 = 4 X 4 X 4). More accurately 

stated, the earth's volume is about 50 times that of the 

moon. 

A somewhat more difficult problem is the " weighing" of 
the moon, which, as we have already seen in the case of 
the earth (p. 103), really means a determination of the 
moon's mass. Curiously enough, the mass of the moon is 
most simply determined by observations of the sun. To 
understand how this is done, we must begin by correcting 
an approximately accurate theory which we have so far 
found sufficient for our explanations. It is frequently 
convenient, for the sake of lucidity, to begin the explanation 
of some phenomenon by assuming a state of affairs resembling 
closely that which actually exists in nature, and afterwards 
substituting new explanations successively, each more 
closely approximating to the truth, until we can finally con- 
sider a tolerably complete theory in its full complexity. 

In the present instance, we must now correct a previous 
statement concerning the earth's orbit around the sun. The 
earth has so far been said to pursue an oval or elliptic orbit, 

173 



ASTRONOMY 

with the sun at one focus. And by the earth we mean, 
of course, the earth's center. But we now know that 
the earth and moon together are traveling in that orbit 
around the sun. Therefore, speaking accurately, it is not 
the earth's center that is exactly in the orbit, but rather 
the combined " center of gravity" of the earth and moon. 
And by center of gravity we mean a point so situated on the 
line joining earth and moon that their weights, as it were, 
would just balance about the center of gravity. Figure 44 
shows this position of the center of gravity at c. If we imag- 
ine the earth and moon attached to the ends of a rigid bar 
240,000 miles long, their weights would balance if the bar 

were supported at the point c. 
_r\ And owing to the great mass 




Moon of the earth as compared with 

Fig. 44. CeDter of Gravity of Earth the moon, this Center of 
and Moon. 

gravity is much nearer the 
earth's center than the moon's. It is, in fact, inside the 
earth's surface. 

Now this center of gravity has another peculiarity of the 
utmost importance. Not only is it the point that is really 
following out the terrestrial orbit around the sun, but it is 
also the real focus about which the moon pursues its monthly 
orbit around the earth. The moon, accurately speaking, 
does not revolve about the earth, but about the point c, the 
center of gravity of the earth and moon combined. Further- 
more, while the moon is going around this center, the 
earth is doing the same thing, though in a much smaller 
orbit. Again imagining both bodies attached to the ends of 
a rigid rod, it is a little as though this rod were pivoted at 
the center of gravity, and turning around it. Thus the 
force of gravitation causes both bodies to revolve 

174 



MOONSHINE 

about their common center of gravity, but little moon can- 
not make big earth travel in as large an orbit as big earth 
imposes on little moon. 

The final result is to swing the earth each half month a 
short distance either forward or backward with respect to 
the position it would occupy in its annual orbit around the 
sun, if there were no moon. Sometimes the earth is in 
advance of the center of gravity; sometimes behind it. 
But we always see the sun projected on the celestial sphere 
at a point on the ecliptic circle directly opposite the 
earth's actual position in its orbit; therefore this center 
of gravity effect must show itself by slightly advancing 
or retarding the sun's apparent motion in the ecliptic 
circle. 

The whole phenomenon is very slight, amounting to a 
total change in the sun's apparent place on the ecliptic 
circle of only 12" of arc. Yet this can be measured with 
accurate instruments ; and a simple calculation then shows 
that the common center of gravity of earth and moon is 
distant only 2880 miles from the earth's center. This is 
about jfe part of the total distance between the centers of 
these two bodies ; therefore the lunar mass must be about 
g\ part of the earth's mass. 1 

Having thus found the moon's volume to be about -fa that 
of the earth, and its mass only ^ T , it follows that the moon 
must on the average be composed of materials less dense 
than those of which the earth is made. If the moon were 
equally dense with the earth, a cubic foot of average lunar 
material would weigh as much as a cubic foot of average 
terrestrial material ; and these ratios of masses and volumes 
between the two bodies would be equal. The figures we have 

1 Note 21, Appendix. 
175 



ASTRONOMY 

obtained make the moon's density about six-tenths that 
of the earth. 

The interval of time between two successive returns of the 
moon to the meridian of any place on the earth may be called 
the Lunar Day. Its length depends on the diurnal axial 
rotation of the earth, in a manner analogous to the relation 
between sidereal and solar days (p. 65). We have seen 
that the sidereal day is equal to the period of the earth's 
axial rotation, and is therefore the interval of time between 
two successive returns of the vernal equinox to the me- 
ridian. We have also seen that the solar day is about four 
minutes longer than the sidereal day, because the sun's 
apparent daily motion of one degree along the ecliptic 
circle makes the sun lag a little behind the equinox point, 
so that the apparent rotation of the heavens must continue 
about four minutes after each complete axial rotation of the 
earth, to enable the sun to reach the meridian again (p. 69). 
The case of the moon is precisely similar ; only, as its daily 
motion averages about 13° instead of 1°, the excess length of 
the lunar day is about 52 minutes, instead of 4 minutes. 
This makes the lunar day average about 24 h 52 m of sidereal 
time. 

The fact that the moon thus reaches the meridian about 
52 minutes later each night means that it will also rise and 
set about 52 minutes later each night. But this is only an 
average figure; in the latitude of New York, for instance, 
the daily retardation of moonrise may vary all the way from 
23 minutes, to 1 hour 17 minutes. 

When this retardation of the time of moonrise is at the 
minimum of 23 minutes, the moon will rise at nearly the 
same time on two or three successive nights. If the 
moon also happens to be almost a full-moon on such an 

176 



MOONSHINE 

occasion, we have the phenomenon known as the Harvest 
Moon. This is defined, then, as the rising of the moon, 
nearly full, on two or three successive nights at nearly the 
same hour. 

To ascertain when this will occur, we must discuss the 
principal cause of these large variations in the daily retarda- 
tion of the time of moonrise. For this purpose we may, with 
sufficiently close approximation, consider the moon as 
appearing always in the ecliptic circle on the sky; as we 
already know, it is actually never very far from that circle. 
This being premised, it is clear that the time-interval be- 
tween the moonrises on two successive nights will depend 
on the angle between the ecliptic 
circle and the horizon, as shown 

in Fig. 45. HH is part of the h ^o h 

horizon ; VV part of the ecliptic ^^^^J' 

circle. Let us suppose the moon 

^^ Fig. 45. Harvest Moon. 

was at the intersection / when 

it rose on a certain night. Exactly twenty-four hours later 
the point I will be again rising above the horizon HH. But 
in those twenty-four hours the moon will have moved along 
the ecliptic to the point I', about 13° from I. 

How much later will the moon rise on the second night ? 
Clearly, by a time-interval exactly equal to the time in which 
the apparent rotation of the celestial sphere will move 
the point V up to the horizon HH. This interval will be 
short, if the angle HIV between the horizon and ecliptic is 
small. 

But the angle HIV is not always the same. It is easy to 

demonstrate, by the aid of a celestial globe, that it is a 

minimum when the point of intersection / is at the vernal 

equinox (p. 35). This is well illustrated by the small photo- 

n 177 




ASTRONOMY 

graph of Plate 6, p. 162. The line joining the moon's horns 
being nearly horizontal, the ecliptic must be nearly perpen- 
dicular to the horizon if the horns are to point directly away 
from the sun (p. 165). And the date of the photograph 
being near the vernal equinox, about the time of sunset, 
it follows that the angle HIV, being nearly a right angle, is at 
a maximum at the western horizon on or about March 21. 
Moreover, near the eastern horizon, it will be at a mini- 
mum on the same date. That the ecliptic rises very high 
from the western horizon at sunset on March 21 is also shown 
by the table on p. 49. 

It results from these considerations that if the full-moon 
occurs when the moon appears near the vernal equinox 
point, the daily retardation of moonrise will be a minimum. 
But we have already found (p. 163) that the full-moon always 
necessarily appears opposite the sun in the sky. Therefore, 
on the occasion of a harvest moon, the sun must be at the 
autumnal equinox (p. 35), which is directly opposite the 
moon's position at the vernal equinox. But the sun appears 
in the autumnal equinox about September 22 in each year. 
Consequently, the harvest moon is always the full-moon 
which happens nearest to September 22. 

And this explains the name " harvest." For certain 
harvests are gathered in September ; and it is of consequence 
to farmers to have plenty of moonlight, so that their work 
may be completed before rain falls. The full-moon, being 
opposite the sun, will rise when the sun sets, which occurs 
at six o'clock on the day of the equinox. Thus the harvest 
full-moon will rise on two or three consecutive dates at about 
six in the evening, and will remain visible until sunrise the 
next morning. 

Still another phenomenon of interest arises from the fact 

178 



MOONSHINE 

that the full-moon always appears opposite the sun in the 
sky. Near the time of the winter solstice (p. 121) in Decem- 
ber the full-moon must be near the summer solstice point 
of the ecliptic circle, in order that it may be opposite the 
sun. It follows from this that the winter full-moons appear 
far north of the celestial equator, like the sun in summer. 
Consequently, the full-moon in winter " rides high," as the 
saying is; when on the meridian it will appear near the 
zenith, while the summer full-moons are low down in the 
sky, like the sun in winter. 

These variations in the time of moonrise are always set 
forth in ordinary almanacs ; but a certain peculiarity of this 
part of the almanacs requires explanation. In the case 
of the sun, the almanacs give both the time of sunrise and sun- 
set, all of which is understood without difficulty. But for 
the moon, the almanacs give only the time of rising or the 
time of setting, — never both. And both are not needed. 
If the moon, for instance, rises shortly after sunset, 
it will set shortly after the next sunrise. It will therefore 
be in the sky when the sun rises, and will set during day- 
light, — a phenomenon not usually observable. In other 
words, only one of the two phenomena, moonrise or moonset, 
can be observed on any given date, and the almanac always 
gives the time of the observable phenomenon. 

But this introduces another complication. As the lunar 
"day" is 24 h 52 m long, it may happen now and then that a 
given solar day of 24 hours contains no moonrise at all. 
The moon might have risen just before the beginning of the 
solar day, and might rise again just after the ending of it. 
In fact, this must occur once each month. If the moon- 
column in the almanac contains the word "rises," the follow- 
ing numbers in the column are the successive times of moon- 

179 



ASTRONOMY 

rise. On the date when the moon does not rise, the abbre- 
viated word "morn" is then substituted in the moon column 
for the usual time of moonrise. The following numbers in 
the column then indicate that the moon rises after midnight, 
— in the morning. 

In the lunar orbit there exists still one more peculiarity 
that illustrates the tendency of astronomy to deceive us by 
entangling the seeming and the true, — a tendency that 
has much to do with the peculiar fascination of the science. 
To an observer on the earth the moon's orbit seems to be 
an ellipse or oval curve ; but the true orbit is not really an 
ellipse at all. For while the moon is traveling around the 
earth, the earth is itself speeding through space in its annual 
orbit around the sun, dragging with it the moon and the 
lunar orbit around the earth. 

Consequently, though the moon's orbit is an ellipse, so 
far as we dwellers on the earth are concerned, its real path in 
space is compounded of the two motions involved : first, 
the lunar motion around the earth ; and second, the terrestrial 
motion around the sun. Now the earth's linear velocity of 
motion around the sun is much more rapid than the lunar 
motion around the earth, and is therefore of greater influence 
in fixing the true shape of the orbit in space. And it is 
known that the sun is itself also moving through space, 
carrying with it the earth and the whole solar system, in- 
cluding the moon. This motion would also affect the shape 
of the lunar orbit ; but we shall here consider only the two 
principal causes already mentioned, — the moon's motion 
around the earth, and the earth's motion around the sun. 

It is a very singular thing, and one not altogether easy 
to understand, that the combination of these motions makes 
the true path of the moon always concave toward the sun, 

180 



MOONSHINE 

as shown in Fig. 46. The arrow indicates the direction of 
the sun ; E h E 2 , E 3 , etc., are five successive positions of the 
earth in its annual orbit around the sun, separated by an 
interval of about 7| days, or one-quarter of a lunar synodic 
period (p. 161). The points M h M 2 , M 3 , etc., are five cor- 
responding positions of the moon. Mi and M 5 are new- 
moon positions ; M 3 a full-moon position ; M 2 and M 4 rep- 
resent quartered phases. The whole line MiM 2 M 3 M^M 5 
represents a part of the moon's actual orbit in space with 
respect to the sun; and we can prove without difficulty 
that it is everywhere concave toward the sun. 1 




Fig. 46. Moon's Path with Respect to the Sun. 

When considering the lunar atmosphere we found the 
moon quite unlike the earth. But there exists also a very 
conspicuous similarity between the two bodies, — the moun- 
tainous character of their surfaces. There are a number 
of mountain ranges on the moon, and numerous craters 
apparently of volcanic origin ; but there are no active 
volcanoes. These lunar mountains are from 1000 to 
2000 feet high, and some of the craters are 50 miles 
in diameter. In the center of the crater there is 
often a conical mountain peak; it is as though the crater 
wall was formed by a shower of volcanic material ejected from 
a center, and falling in a circle around it. The central 
peak may then have resulted from a final outburst of the 
volcanic discharge, after the explosive force of the volcano 

1 Note 22, Appendix. 
181 



ASTRONOMY 

had become too feeble to throw its lava far from the eruptive 
center. The moon's surface also shows many " rills" or 
crooked valleys radiating from certain craters. These 
surface features are well seen in Plate 7. At the bottom of 
this photograph is the great crater Theophilus, with its 
rugged central mountain peak. 

The height of lunar mountains and crater walls may be 
measured with the telescope. In certain lunar phases, when 
sunlight falls obliquely on the moon's surface, the moun- 
tains cast long black shadows, seen conspicuously in Plate 7. 
It is possible to measure in the telescope the angular length 
of such shadows ; and knowing the moon's distance, we can 
then calculate the shadow lengths in miles from the measured 
angular lengths. (Cf. p. 172.) Then, from the calculable 
angle at which sunlight falls on the lunar surface at the 
moment when the shadows were measured in the telescope, 
and the known shadow lengths in miles, we can compute 
the mountain heights, also in miles, by methods well known 
to surveyors. 



182 




PLATE 7. Lunar Enlargement. 



Photo by Ritchey. 



CHAPTER XI 

THE PLANETS 

In discussing the celestial sphere (p. 23) and the ap- 
pearance of the stars projected upon it, we found that the 
great mass of these luminous points retain practically 
unchanging relative positions on the sphere, and are subject 
only to such apparent motions as result from the earth's 
daily rotation on its axis and annual orbital revolution 
around the sun. At the same time, a certain small number 
of stars move about among their fellows (p. 10). These 
are the " wanderers," — the Planets. Five are easily visible 
to the unaided eye, — Mercury, Venus, Mars, Jupiter, and 
Saturn. Uranus may also be seen without a telescope under 
favorable conditions ; Neptune, and the great body of tiny 
telescopic objects of the planetary class, called Planetoids, 
require optical help to be seen. 

The distinguishing thing about these planets is that 
they all belong to our solar system. The earth is merely 
one of the planets in that system ; the others, like the earth, 
revolve around the sun in orbits analogous to the earth's 
own annual orbit. These planetary orbits are all oval 
or elliptic, and have the sun at a point near the center of 
the orbit, — the focus (p. 116). 

When explaining the earth's annual orbital revolution 
around the sun, we described a simple method of observation 
by which the form of the terrestrial orbit might be deter- 
mined experimentally. These simple observations were also 

183 



ASTRONOMY 




Fig. 47. Law of Areas. 



found capable of establishing for the earth a law of planetary 

orbital motion first discovered by Kepler; viz. that the 

"radius vector" (p. 119), or line joining the planet and the 

sun, moves over equal areas in 
equal times. Thus, in Fig. 47, S 
represents the sun, P h P 2 , P 3 , P 4 , 
four positions of a planet in its 
orbit, such that the motion from 
Pi to P 2 is accomplished in the 
same interval of time required for 
motion from P 3 to P 4 . Then the 
triangular area SPiP 2 , included be 
tween the two radii vectores SP h 

SP 2 , and the arc of the curved orbit PiP 2 , is equal to the 

other triangular area SP3P4, similarly included between two 

radii vectores and an arc of the curved orbit. 

We must now prove that this law applies universally to all 

planets, and that it is a necessary consequence of Newton's 

law of gravitation. This latter law, as we have already 

seen (p. 103), declares that an attraction 

exists between the sun and planet, directly 

proportional to the product of their masses, 

and inversely proportional to the square of 

the distance between them. 

Let us consider Fig. 48, and suppose 

that at a certain instant of time the sun 

is situated at S, with the planet at Pi ; 

and let us first examine what the planet's 

motion would be if there were no such 

thing as an attraction toward the sun. We may suppose 

the planet to be traveling with a certain velocity, and in a 

certain direction, such as would carry it to P 2 at the end of 

184 




Fig. 48. Planetary 
Motion. 



THE PLANETS 

one second of time. This original velocity and motion may 
be regarded, if we choose, as a result of the original cata- 
clysm whereby the planet was first brought into separate 
existence. 

Now if there were no attraction toward the sun, and as there 
can be no friction or resistance to motion in empty space, 
the planet will arrive at P 2 endowed with the same velocity 
and direction of motion which it originally possessed at Pi. 
Therefore it will, under these circumstances, travel an equal 
distance along the same straight line in the next second, 
and thus arrive at P 3 . The line P1P2P3 is the planet's 
orbit, if there be no attraction toward the sun, and the 
lines SPi, SP 2 , SP 3 , are three positions of the planet's radius 
vector. 

The area traveled over by the radius vector in the first 
second is the triangle SPiP 2 ; and in the second second it 
is the triangle SP 2 P Z . But these two triangles have equal 
areas; 1 and this constitutes a proof that the radius vector 
moves over equal areas in equal times, if there exists no 
attraction whatever toward the sun. 

Next suppose that the solar attraction exists, but that 
instead of being continuous in action it is applied suddenly 
in the form of an impulse toward the sun at the end of each 
second of time. Suppose the first impulse is applied at the 
end of the first second of time, when the planet has reached 
P 2 , and that it is applied toward the sun along the radius 
vector P 2 S. Now consider Fig. 49, and imagine the im- 
pulse toward the sun strong enough to have carried the 
planet to P 2 ' in one second of time, supposing the said im- 

1 Readers familiar with geometry will recognize that these triangles 
are equal because they have a common vertex at S, and equal bases 
P1P2 and P 2 P 3 situated upon a single straight line. 

185 



ASTRONOMY 

pulse toward the sun to have acted alone during one second. 
But the planet at P 2 is also subject to the original force, 
which, acting alone, would have moved it to P 3 in the second 
second of time. Thus the planet at P 2 is subject to two 
forces, one of which, acting alone, would have carried it to 
P 2 ' in the next second ; and the other, likewise acting alone, 
would have carried it to P 3 in that next second. 

Where will the planet go under the combined action of 
these two forces in the second second of time? It must 
evidently move along a line intermediate 
in direction between P2P3 and P2P2'. 
That line will in fact be PiP*, and at 
the end of the second second the planet 
will arrive at the point P3'. 1 Its radius 
vector will then be the line SP S ' ; and 
the areas traversed by the radius vector 
in the two consecutive seconds of time 
here under consideration will be the tri- 
angles SPiP 2 and SP2P3. It is not difficult to show that 
the areas of these two triangles are also equal. 2 Conse- 
quently, under our present supposition as to the nature of 
the attraction toward the sun, the planetary orbit PiP 2 P 3 ' 
still satisfies the law of areas. 

It is evident that any number of impulses toward the sun 
at the ends of other successive seconds of time would pro- 
duce similar results. And the same reasoning would hold 
true if we suppose the impulses to occur more frequently; 
say ten or a hundred times in a second of time. It follows 
that if we increase sufficiently the number of supposed im- 
pulses per second, we can at last transform our orbit from a 
series of very short straight lines into an actual curve; for 
1 Note 23, Appendix. 2 Note 24, Appendix. 

186 




THE PLANETS 

every curve may be regarded as made up of an infinite 
number of excessively short straight line elements. And 
at the same time, the supposed series of impulses toward 
the sun, coming infinitely close together, are transformed 
into the continuous action of gravitational attraction. 
The above reasoning therefore constitutes a proof that a 
planet moving under the influence of an original impulse 
in any direction, plus a gravitational attraction toward 
the sun, will pursue an orbit satisfying the law of equal 
areas for the radius vector. 

One of the most interesting things in the above proof is 
the absence of any special requirements as to the nature of 
solar gravitational attraction. Nothing in the proof de- 
mands an attraction acting accurately in accordance with 
Newton's law (p. 103). To satisfy the law of areas, it is 
merely necessary that the attracting force be what is called 
a " central" force, directed always toward a definite point 
occupied by the sun within the orbit. And conversely, the 
fact that the planets can be observed to travel in orbits that 
satisfy the law of areas, proves merely that they are moving 
under the influence of a central force, but not necessarily 
that particular variety of central force which we know under 
the name of Newtonian gravitation. 

But in addition to this law of areas, which can be deduced 
as a fact directly from observation (p. 120), two other 
similar laws are known, — also obtainable directly from 
observation. All three laws were first found by Kepler; 
they are called, to the present day, Kepler's three laws 
of planetary motion; and they may be formulated as 
follows : 

1. The orbit of each planet is an ellipse, with the sun at 
the focus of the curve. 

187 



. ASTRONOMY 

2. The radius vector of each planet passes over equal 
areas in equal times. 

3. If the time required by any planet to complete a revolu- 
tion in its orbit is called its "period," then the squares of the 
planetary periods are proportional to the cubes of their 
average distances from the sun. This third law is called 
the "harmonic law." 1 

We have just proved that the second law, or law of areas, 
is a necessary consequence of the existence of a central 
force pulling always toward the sun. It is similarly possible 
to prove, by the aid of mathematics, that all three laws 
follow as a necessary consequence of a central attracting 
force, provided that force acts in accordance with the 
Newtonian law. Thus the three laws of Kepler are merely 
corollaries or consequences of Newton's more general law; 
Newton's great service consisted in bringing everything under 
the sway of a single law, instead of three separate ones, 
apparently unrelated. 2 

In the light of the above explanation of Kepler's and 
Newton's work, it will now be of interest to give a brief 
account of the two best known explanations of planetary 
motion within the solar system, — the Copernican theory, 
which, with some modifications, is the one now accepted, and 
the older Ptolemaic theory. It may possibly seem out of 
place to give any attention to the abandoned Ptolemaic 
hypothesis; it is like studying something we know to be 
untrue. But there are many references to that theory in 

1 The harmonic law may be represented mathematically by a simple 
proportion : 

Let h, t 2 , be the periods of two planets, a Y , a 2 , their mean distances from 
the sun. 

Then : tl 2 . U 2 = 0l s . fl2 3. 

2 Note 25, Appendix. 

188 



THE PLANETS 



Saturn 



literature : a few pages may well be devoted to a description 
of it ; enough, at least, to form some idea of its peculiarities. 
It is also of interest that the Ptolemaic theory was actually 
taught in early days at Harvard and Yale colleges, as being 
a possible alternative theory to 
the Copernican. 1 

Ptolemy (140 a.d.), following 
Hipparchus, supposed the earth 
to be immobile, near the center 
of the universe. For each planet 
a circular orbit was provided 
(Fig. 50), which circle was called 
the planet's " deferent." Upon 
the deferent moved, not the 
planet itself, but an imaginary 
planet, represented by a point. 
The actual planet moved in 
another circle called the " epi- 
cycle," whose moving center was 
the imaginary planet. The sun 
and moon had deferents, but no 
epicycles. Each deferent was 
supposed to be traced on the 
surface of a perfectly trans- 
parent separate crystal sphere ; 2 
and all these crystal spheres rotated once a day around 
an axis passing through the poles of the heavens. The 
outermost crystal sphere had no deferent or attached 
epicycle ; but to it were fastened all the fixed stars. This 




Fig. 50. Ptolemaic Theory. 



1 Young, Manual of Astronomy, p. 323. 

2 These spheres, by their motion, produced the famous "music of the 
spheres." 

189 



ASTRONOMY 

star-sphere also rotated around the polar axis of the 
heavens. 

The spheres being all of crystal, and perfectly transparent, 
did not interfere with a view of what was going on outside of 
each in connection with the exterior deferents and epicycles. 
The daily axial rotation of the spheres produced all the 
diurnal phenomena we now believe to result from the 
axial rotation of our earth. And the spheres, of course, 
revolved from east to west, not as our esCrth does, from west 
to east. 

The deferents of Mercury and Venus were inside the solar 
deferent. The imaginary planets Mercury and Venus 
revolved in their deferents once a year, keeping pace with 
the solar motion in its own deferent circle. The sun and the 
two imaginary planets Mercury and Venus were always in 
line, as shown in the figure. The revolution of the actual 
planets Mercury and Venus in their epicycles thus made 
them swing back and forth, east and west of the sun, in a 
manner quite similar to their actual observable apparent 
motions to be described later in the present chapter. 

Mars, Jupiter, and Saturn were connected by Ptolemy to 
deferent circles exterior to the sun. The periods of revolu- 
tion of the imaginary planets were not here assumed equal to 
that of the sun, as was the case for the inferior planets 
Mercury and Venus ; and in this way the observable phe- 
nomena were also reproduced for these superior planets. 
Later investigators, following the Ptolemaic theory, added 
further secondary imaginary planets, revolving in Ptolemy's 
epicyclic circles ; with the actual planets attached to addi- 
tional corresponding epicycles. In this way they were able 
to reproduce all irregularities of motion, as improving 
methods of observing brought them to light. 

190 



THE PLANETS 




In contradistinction to the above, the Copernican theory, 
as we have already seen, supposes the sun immobile, and the 
planets moving in flattened oval orbits with the sun at one 
focus. The great objection to this system, an objection that 
long prevented its adoption by men of science, is this : if 

the earth really 
revolves in an orbit 
around the sun, the 
fixed stars should 
change their ap- 
parent positions, 
as seen projected on the sky, while the 
earth progresses around its orbit. Figure 
51 makes this clear. Let S be the sun; 
W and E" two positions of the earth at 
opposite points of its orbit. Suppose a 
star to be situated in space at P, fixed and 
immobile. Then from E' we should see 
the star projected on the celestial sphere 
at P', and from E" we should see it at P". 
As a matter of fact we see each fixed star 
constantly projected in the same place on 
the celestial sphere; and this seemed an 
insuperable objection to many early astron- 
omers, including the famous Tycho Brahe. 
Fig. 5i. Copernican On the other hand, if the earth does not 
move, there would of course be no change 
in the direction of the lines E'P' and E"P" . There would 
be but one such line if the earth were constantly in the 
center at S. 

This objection to the Copernican system was not removed 
until the middle of the nineteenth century, when, for the 

191 



ASTRONOMY 

first time, Bessel was able to measure with certainty a slight 
difference between the two sight lines from the earth to a 
certain star in the constellation Cygnus. It then appeared 
that the trouble arises from the extreme minuteness of the 
angle E'PE", caused by the fact that the fixed stars are all 
so excessively distant, in comparison with the diameter of the 
earth's orbit. And, of course, the angle E'PE" will diminish 
with an increasing distance of the stars. Up to the present 
time, no star has been found for which this angle exceeds 1.5 
seconds of arc ; and in the case of but very few stars has the 
angle been found large enough to be measured, even with 
the powerful astronomical instruments of to-day. The 
angle subtended at P by the radius of the earth's orbit is of 
course half the angle E'PE". This half-angle is called the 
star's " parallax." And the measurement of even a single 
stellar parallax removes the fundamental difficulty of the 
Copernican theory. 

Of historic importance even greater than the above 
theory of Ptolemy are certain very old and very simple 
methods of determining observationally a planet's pe- 
riod of revolution around the sun and distance from the 
sun in terms of the earth's distance. It is evident that 
before Kepler discovered his harmonic law, no relation was 
known to exist between distance and period; but there 
were always simple methods for determining the period by 
direct observation. When we were discussing the earth in its 
relation to the sun (Chapter VII), we found that the great 
ecliptic circle on the sky is cut out by the plane of the ter- 
restrial orbit produced outward to infinity. It must also 
be a fact that any planetary orbit plane cuts the ecliptic plane 
in a straight line, because any two planes in space must in- 
tersect in a straight line. This intersection line is called 

192 



THE PLANETS 

the line of nodes of the orbit. Twice in the course of its 
revolution around the sun the planet must reach this line 
of nodes. When this occurs, the planet is for a moment in 
the ecliptic plane as well as in the plane of its own orbit ; 
and as the earth is always in the ecliptic plane too, it follows 
that, at this critical moment, the straight line joining the 
earth and planet will lie entirely in the ecliptic plane. 

But we see the planet along that line, observing from the 
earth toward the planet. Consequently, if we observe 
the planet at the critical moment, we shall see it projected 
on the sky somewhere in the great circle cut out on the sky 
by the ecliptic plane. So we can ascertain by observation 
when the planet is in the node, by noting the instant of time 
when it crosses the ecliptic circle, as seen projected on the 
sky. The interval between two successive passages of the 
planet through the same node is then its period of revolution 
around the sun. 

Kepler made certain important improvements in the 
above method of determining planetary periods ; and, of 
course, he also gave much time to the study of planetary dis- 
tances from the sun, in the work preparatory to his discovery 
of the three great laws. As an example of Kepler's ingenious 
methods, we shall give here his investigation of the varia- 
tions in the distance between the earth and the sun in dif- 
ferent parts of the terrestrial orbit. 1 Kepler had at his 

1 Kepler's works are in Latin, and are difficult to read. The original 
book from which we quote in modernized form is called " Astronomia Nova 
seu physica coelistis tradita commentariis de motibus stellae Martis ex obser- 
vationibus Tychonis Brake." It was published in 1609, but there is a 
reprint by Dr. Charles Frisch, published in 1860 "Frankofurti et Er- 
langae." 

A most excellent commentary on Kepler was also published in London 
in 1804 by the Reverend Dr. Robert Small, and dedicated to the Earl of 
Lauderdale. 

o 193 



ASTRONOMY 

disposal a long series of observations of the planet Mars, 
accumulated by his master Tycho Brahe. These observa- 
tions recorded the positions of Mars as seen projected on the 
sky on a very large number of different dates. He selected 
certain of these observations dated as follows : 1 

1590, March 5, 7 h 10 m , 

1592, Jan. 21, 6 h 41 m , 

1593, Dec. 8, 6 h 12 m , 
1595, Oct. 26, 5 h 44 m . 

It will be seen that he had been able to choose four ob- 
servations separated by exactly the same interval of time ; 

viz. : 686 d 23 h 31 m , which 
interval corresponds very 
nearly with the known 
average period in which 
Mars completes a revolu- 
tion around the sun. In 
the accompanying Fig. 
52, therefore, Mars must 
occupy the same position 
M on each of the above 
dates, while the earth will occupy the successive positions 
E, F, G, H. These terrestrial positions will be equidistant 
points on a circle with its center at B, if we suppose 
that the earth moves uniformly in a circular orbit. Under 
this supposition, these points must be equidistant, since 
they are separated by a series of equal time-intervals, 
each equal to the Martian period. And it is important to 
notice that the successive returns of Mars to the same point 
M are independent of any assumption as to the form or 

1 Frisch, Kepler, Vol. 3, p. 275 ; Small, Kepler, p. 202. 
194 




Fig. 52. Kepler's Mars Observations. 



THE PLANETS 

position of the Martian orbit. Whatever and wherever this 
orbit may be, Mars must return to the same point after 
each complete orbital revolution has been terminated. 

For the date 1590, Mar. 5, when the earth was at E, Kepler 
had Tycho's observation of the position of Mars as pro- 
jected on the ecliptic circle, or rather the position of that 
point on the ecliptic circle which was nearest to Mars. This 
gave the direction of the sight-line EM from the earth to 
Mars. The directions of the lines from the center B to the 
earth, and from the center to Mars, were furnished by the 
tables of planetary motion in Tycho's possession. Thus 
the directions of the three sides of the triangle EBM were 
known, and from these the three angles of the triangle were 
obtained by subtraction. 

But when the three angles of a triangle are known, it is 
possible to calculate the relative lengths of the triangle's 
sides. 1 By successive applications of this process, Kepler 
computed that 2 — 

BE = .66774 X BM. 
BF = .67467 X BM. 
BG = .67794 X BM. 
BE = .67478 X BM. 

These numbers should all be equal if the point around 
which the earth describes equal angles in equal times were at 
B, the center of the circle in which the earth is supposed to 
move. So Kepler's numbers show that the point about 
which the earth's angular motion is uniform is not at the 
center B of the earth's orbit, supposed circular ; but that it is 
at some point C, outside the center of the orbit. Kepler was 

1 In the language of trigonometry, the sides are proportional to the 
sines of the opposite angles. 

2 Frisch, Kepler, Vol. 3, p. 275. 

195 



ASTRONOMY 

able to compute the position of this point C ; and the corre- 
sponding changing distance between the earth and the sun. 
These results were of course obtained long before he perfected 
his three laws ; they are regarded justly as marking one of 
the most difficult and important advances ever made in 
human knowledge. 

There is still another remarkable peculiarity about the 
planetary distances from the sun; like the foregoing, of 
historic interest only. When we compare these distances, 
we find an accidental relation between them. Let us number 
the planets consecutively, from the sun outward, calling 
Mercury, 1 ; Venus, 2 ; Earth, 3 ; Mars, 4 ; the Planetoids, 
5 ; Jupiter, 6 ; Saturn, 7 ; Uranus, 8 ; Neptune, 9. Let us 
then multiply the number 2 by itself four times, say for 
Mars, which is planet number 4. This gives 16. Then 
take three-quarters of this number, giving 12. Increase 
this result by 4, giving 16. Divide this by 10, giving 1.6. 
The result is an approximate value for the distance of 
Mars from the sun, counting the earth's distance from the 
sun as 1. 

This curious arbitrary rule is known as Bode's law; 
astronomers have been acquainted with it for more than a 
century ; but we know of no physical reason why it should 
have a real existence. The following little table contains a 
comparison of the known planetary distances with their 
values calculated as above. In the case of the planetoids 
an average value is given. 

The table shows that the law is quite accurate until we 
reach Neptune ; then the error increases suddenly ; and we 
must conclude that the whole thing is one of those rare and 
remarkable coincidences that nature sometimes provides, 
apparently to mislead scientific investigators. 

196 



THE PLANETS 



Planet 


No. 


Known 
Distance 


" Bode" 
Distance 


Error 
" Bode" 


Mercury 

Venus 

Earth 

Mars 

Planetoids .... 
Jupiter . . . . . 
Saturn , . . 

Uranus 

Neptune 


1 
2 
3 
4 
5 
6 
7 
8 
9 


0.4 
0.7 
1.0 
1.5 
2.6 
5.2 
9.5 
19.2 
30.0 


0.5 

0.7 

1.0 

1.6 

2.8 

5.2 

10.0 

19.6 

38.8 


0.1 
0.0 
0.0 
0.1 
0.2 
0.0 
0.5 
0.4 
8.8 



Having thus described certain famous historic methods 
of studying the planetary distances, etc., we shall next 
give a somewhat more detailed description of the planetary 
orbits, and the exact nature of the observations by means 
of which we study them in modern times. When we deter- 
mine the position of a planet by observation, we really deter- 
mine only the direction in which we see it projected on 
the celestial sphere. We point a telescope at the planet, 
and, by moving the telescope, bring the center of the plane- 
tary disk very accurately into the middle point of the field 
of view, which, for this purpose, may be supposed to be 
fitted with a very fine pair of cross threads to mark the center. 
Then, if the telescope mounting be provided with suitable 
"graduated" x circles, we can read the angles measured by 
those circles, and thus ascertain the direction of the planet 
in space, referred to certain points and lines, such as the 
celestial poles and equator. In other words, we measure 
the planet's right-ascension and declination (p. 34), as it is 
seen projected on the celestial sphere. 

We can also note the exact time when this observation 

1 Brass circles divided into degrees, minutes, and seconds of arc. 

197 



ASTRONOMY 

was made, thus fixing the moment when the planet's direc- 
tion from the earth was measured. There are other methods 
of making these observations in addition to direct measure- 
ment with graduated circles attached to the telescope ; but 
all are alike in this : they furnish us with the direction in 
space of the sight-line joining the earth with the planet, 
and the instant of time when that line had the direction 
in question. Direct observation gives no information 
whatever as to the planet's distance from the earth. It tells 
us nothing about the length of the line joining earth and 
planet ; only its direction in space. 

If several observations of this kind have been made at 
different times, separated perhaps by a number of days, 
or even months, the earth will itself have moved consider- 
ably in its own orbit in the interval between the observa- 
tions. The planet will also have moved in its own orbit. 
Consequently, both ends of the line will have moved in 
different orbits and with different velocities; so that the 
changes in direction of the line will have been of an extremely 
complicated nature. 

But the changes in space of one end of the line are well 
known to us, — the earth end. For we know the orbit of 
the earth around the sun, and can calculate the terrestrial 
position in space accurately for each moment of time when 
an observation was made. Knowing thus, from calculation, 
the position of one end of the line, and, from observation 
the direction of the line, the line itself becomes fully known, 
all but its length. Thus, in Fig. 53, if at a certain time t\ 
the earth was at a known point of its orbit E h and the 
planet was seen in the observed direction Pi, we know the 
line E1P1, all but its length. If, at a second observation, 
made at the time t 2 , the earth was at E 2 , and the planet 

198 



THE PLANETS 

was seen in the direction P 2 , we again know the line E0P2, 
excepting its length. And the same is true of a third line 
E Z P Z . But it is to be remarked that these three lines 
will not lie in a single plane, unless the terrestrial and plane- 
tary orbits around the sun should happen to lie in the same 
plane, which is not accurately the case in our solar system. 

The problem now is to determine the planetary orbit 
from observations of this kind. But we know certain 
additional things about this planetary orbit. We know 
that it is an ellipse or oval; that the 
sun is in the focus ; and we know the 
position of the sun with respect to the 
earth from our knowledge of the terres- 
trial orbit, since the sun is also in the 
focus of that orbit. Both orbits have 
the same point for a focus, and the sun 
is in that point. Furthermore, we 
know the planet's orbital motion must / """"— »-p, 

be such as to satisfy Kepler's laws of FlG - 53 va ^ s et 0bser ~ 
planetary motion (p. 187), and so we 
know the planet must have moved in such a way as to 
cause its radius vector to sweep over areas proportional 
to the known time-intervals between the observations. 

It is a fact that an unknown planetary orbit can be thus 
determined from three observations such as have been de- 
scribed. Our geometric problem may therefore be stated 
thus : 

Given three observed straight lines in space; it is re- 
quired to find an ellipse, cutting these three lines in three 
points, such that the radii vectores to the sun or focus from 
these three points will satisfy Kepler's laws. 

It would carry us too far afield in mathematical astronomy 

199 




ASTRONOMY 

to deduce here the methods by which this problem can be 
solved ; but several interesting things about it can be 
enumerated. In the first place, the problem always has 
two solutions : there are always two ellipses in space that 
satisfy the problem. One of these is the planetary orbit ; 
the other is the earth's own orbit. For the latter is also 
an ellipse; it cuts the three lines because they are sight 
lines from the earth to the planet ; and the earth's motions 
in its own elliptic orbit, of course, satisfy Kepler's law. 

But suppose the problem to have been solved for the 
planet, also ; let us see what we need to know about the 
orbit in order to say that we know the orbit completely. 
Six different things must become known, and six only; 
these are called the six " elements" of a planetary orbit. 

First we must know the length of the largest diameter 
of the ellipse, and the degree of flattening, — the eccentricity, 
as it is called. These two elements being known, we know 
the size and shape of the orbit. We could draw it to scale. 

Next we must know two more things, to define where the 
orbit is located in space. These two elements fix or deter- 
mine the position in space of the plane in which the orbit 
lies. To fix this plane, we must know the angle it makes 
with the plane of the earth's orbit, the ecliptic plane; and 
we must know the position in the ecliptic plane of the line 
along which the planetary orbit plane cuts the ecliptic 
plane. This, as we have seen, is called the "line of nodes" ; 
and the angle between the two planes is called the " inclina- 
tion" of the planet's orbit. 

Having thus fixed the size and shape of the orbit in its 
plane, and the position of the plane itself in space, we must 
still know two more elements. We must know where the 
planet was in its orbit at some definite time ; and we must 

200 



THE PLANETS 

know the position of the orbit in its own plane. As we 
have already seen (p. 120) the planet is said to be in peri- 
helion when it is so placed in its orbit as to be at its nearest 
possible approach to the sun. The perihelion point is that 
point of the orbit which is nearest the sun. Therefore we 
use for one of the orbital elements the exact time of peri- 
helion passage. This element fixes the position in the orbit 
occupied by the planet at a definite moment of time. Finally, 
to locate the orbit in its own plane, we must know the 
direction in that plane of the " major axis," or longest 
diameter of the oval orbit. 

The six elements of a planetary orbit are therefore the 
following : 

1. Longest diameter of oval 1 These fix the size and shape of 

2. Eccentricity, or degree of flattening J the orbit. 

3. Inclination of orbit plane 1 These fix the position in space of the orbit 

4. Position of lines of nodes J plane. 

5. Time of perihelion passage. 

6. Direction of orbital major axis in its own plane. 

A seventh orbital element is usually added : the Period, 
or time required for a complete orbital revolution of the 
planet. But this element is not really an independent one ; 
for the planetary periods and the diameters of the orbits 
are connected by Kepler's harmonic law (p. 188), by means 
of which either may be calculated from the other. 

The elements of an orbit once computed from three com- 
plete observations of the planet's apparent position, as pro- 
jected on the celestial sphere, and seen from the earth, the 
problem can be inverted, and the subsequent apparent pro- 
jected positions of the planet calculated from the elements. 
Thus is it possible to predict exactly where each planet may 
be seen in the sky. If a series of such calculated predictions 

201 



ASTRONOMY 

are tabulated for every day in the year, the tabulation is 
called a planetary Ephemeris. The United States govern- 
ment publishes such a tabulation annually, under the title 
"The American Ephemeris and Nautical Almanac. " In it 
the planetary positions are printed for each day in the 
form of right-ascensions and declinations; and by means 
of these printed numbers it is easy to find the planets in 
the sky. 

The measurement of a planet's axial rotation period, cor- 
responding to the terrestrial sidereal day, is not a very easy 
matter. The best method of doing it is to observe with 
the telescope any spot or mark that may be distinctly visible 
on the planet's surface. As the planet turns on its axis, this 
spot will alternately appear and disappear; for it will of 
course be invisible when the planet's rotation carries it 
around to the side which is turned away from our earth. 
If we note the exact time elapsing between two successive 
returns of the spot to the apparent center of the planet's 
disk, this interval will be the planet's rotation period, or day. 

Such an observation must, of course, be corrected for any 
effects produced by variations in the relative positions of the 
planet and the earth, due to their respective orbital motions. 
And the result can also be much improved by allowing a 
considerable number of rotations to elapse between the two 
observations. If this can be done, the effect of any error 
in noting the exact time when the spot arrives at the center 
of the disk will be greatly diminished. But none of the 
planets, with the exception of Mars, have spots sufficiently 
perfect to admit of precise observation. Our knowledge as 
to the duration of the planetary days is therefore still very 
defective. 

If the paths of the spots, as they move across the visible 

202 



THE PLANETS 

planetary disk, can be mapped with sufficient accuracy, we 
can further ascertain from them the location of the plane- 
tary rotation poles, the inclination of the planetary equator 
to the plane of the orbit, and other related matters. Unfor- 
tunately, information of this kind is still very meager, on 
account of the lack of suitable spots on the surfaces of most 
planets. 

We shall next consider the measurement of a planet's 
size, its diameter, surface area, and volume. We have seen 
that ordinary astronomical instruments enable us to measure 




Fig. 54. Planetary Diameter. 



only the directions in which we observe the heavenly bodies 
projected on the celestial sphere. Thus, for instance, we 
can determine whether a star lies in the direction of the 
celestial equator, or whether its direction makes an angle 
of 10° with the direction of the celestial equator. If the 
former, the declination (p. 34) of the star would be 0°; if 
the latter, 10°. 

Now if we thus observe the difference in direction of the 
two sides of a planetary disk (pp. 13, 52), we have at once 
the " angular diameter," or the angle subtended by the 
planet to an observer on the earth. Figure 54 explains this 
matter. E is the observer on the earth, P the disk of the 
planet. The two arrows show the directions in which the 
observer sees the two sides of the planetary disk projected 

203 



ASTRONOMY 

on the celestial sphere. The small angle at E is the differ- 
ence of these two directions, and it is the angular diameter 
of the planet, which is measured by observation. 

A knowledge of this angular diameter tells us nothing 
about the actual diameter of the planet in miles, unless we 
know also the distance D between the earth and the planet. 
For it is obvious that it would require twice as big a planet 
to subtend the angular diameter observed at E, if the planet 
were removed to double the distance D. But the distance D, 
at the moment of observation, can always be calculated, if 
we know the dimensions and other particulars of the orbits 
pursued by the earth and the planet around the sun. And 
with the distance D available, it is easy to calculate the 
planet's diameter in miles from the observed angular diam- 
eter. 1 

Having thus found the planet's diameter in miles, it is 
frequently convenient to represent it in terms of the earth's 
diameter as a unit. We can then find the surface area of 
the planet, as compared with that of the earth, by simply 
squaring the planet's diameter expressed in terms of the 
earth's diameter as unity. And the same number cubed 
will give us the planet's volume, as compared with the 
earth's. For it is a well-known mathematical principle that 
the areas of spherical bodies are proportional to the squares 
of their diameters ; and their volumes are proportional to 
the cubes of the diameters. 

A somewhat more difficult problem is the determination 
of a planet's mass. If there happens to be a satellite re- 
volving around the planet, the problem is comparatively 

1 This involves merely a trigonometric solution of the long, narrow 
triangle shown in Fig. 54, using the angle at E, which has been measured, 
and the two including sides, which are both equal to D in length. 

204 



THE PLANETS 

easy. We can then determine by observation the period of 
the satellite's revolution in its orbit around the planet ; and 
its distance from the planet in miles can also be observed 
by precisely the same process just used to ascertain the 
planet's own diameter in miles. From these data the 
planet's mass can be computed. 1 

With regard to the planet's satellites in general, there is 
not much more to be said. Their distances from the 
planets are determined, as we have just seen, by means of 
angular measures. Their periods of revolution around the 
planets are best found by noting the time elapsing between 
successive " elongations," or occasions when the satellite's 
orbital motion around its planet carries it to its greatest 
apparent angular distance from the planet. 

Most satellite orbits are almost exact circles : our own 
moon has an exceptionally flattened or elliptic one. And 
the planes of the satellite orbits are mostly very near the 
planes of the planets' equators ; indeed, the equatorial bulg- 
ing of the planet itself should suffice to pull the orbit plane 
of a close satellite into the planetary equatorial plane, from 
gravitational causes alone. That the planets have an in- 
creased diameter at the equator, and a corresponding polar 
flattening, has been verified by direct measurements in the 
case of our earth (p. 97). For the other planets its exist- 
ence is proved by comparing separate determinations of 
polar and equatorial angular diameters, if the position of 
the poles has become known. When the satellites are un- 
usually far from their planets, as in the case of our moon, 
their orbits lie nearly in the planes of the planets' own 
orbits around the sun. 

Before leaving this subject of orbits in the solar system, 

1 Note 26, Appendix. 
205 



ASTRONOMY 

we shall discuss briefly the permanence or " stability " of 
those orbits. Will they endure forever? Will the solar 
system change materially in the course of time ? 

The planets move primarily under the influence of solar 
attraction as if they were themselves mere particles devoid 
of more than an infinitesimal mass. They are, in fact, all 
extremely small in comparison with the great sun. Never- 
theless, they do possess mass in a certain degree ; and con- 
sequently there is an interaction between them, which 
shows itself in slight perturbative effects upon the planetary 
orbits. In other words, if, by any method, we determine 
the elements of a planetary orbit in any given year, we shall 
not find these elements remaining unchanged forever. 
After the lapse of sufficient centuries, the planetary inter- 
actions and perturbations effect changes in the orbital ele- 
ments of the solar system. 

These changes are of two kinds : 

1. The Periodic perturbations. 

2. The Secular perturbations. 

The periodic perturbations increase and diminish in com- 
paratively brief intervals of time, comparable in length to 
the orbital periods of the planets themselves. But the 
secular changes, produced, as it were, in each orbit by all 
the other orbits acting upon it, are extremely slow in period, 
requiring many thousands of years to complete a cycle. 

The periodic perturbations never displace the position 
in which we see a planet projected on the celestial sphere 
more than about one or two minutes of arc, except in the 
case of Jupiter and Saturn, which are at times displaced 
from their proper or unperturbed orbital positions as much 
as half a degree, more or less. 

The most interesting facts about the secular perturbations, 

206 



THE PLANETS 

known from the researches of Laplace and Lagrange, are as 
follows : 

1. The major diameters and periods of the orbits do not 
change. 

2. The inclinations and eccentricities vary in an oscilla- 
tory manner. 

3. The nodal points and perihelion points move around 
the ecliptic and orbital planes, respectively. 

4. All changes of whatever kind are probably oscillatory ; 
so that the solar system is stable and permanent. After 
the lapse of sufficient ages, it will always return again to its 
original condition, no matter what changes it may have 
undergone. Of this, however, there exists a slight doubt, 
due to a possible imperfection discovered recently in La- 
place's mathematical demonstrations. 

5. There is in the solar system an " invariable plane," 
not subject to change, and containing the center of gravity 
of all the bodies composing the system. 

Throughout the foregoing explanations, the word " period" 
has been used to indicate the interval of time required by a 
planet to complete an orbital revolution around the sun. 
But there exists more than one kind of planetary period. 
When we were discussing the planet earth, the sidereal 
year (p. 128) was defined as the time required by the earth 
to complete one orbital revolution around the sun. Thus, 
if we imagine an observer situated on the sun, the sidereal 
year will be the time elapsing between two successive ap- 
parent returns of the earth to the same fixed star, if both 
star and earth are supposed to be seen from the sun, and 
projected on the celestial sphere. In the same way, the 
sidereal period of any planet is the time required for a com- 
plete orbital revolution, from any fixed star back to the 

207 



ASTRONOMY 

same star, and seen from the sun. So far as the sidereal 
period is concerned, then, the earth is in precisely the same 
condition as all the other planets. 

We also found (p. 128) that the earth has a tropical year, 
used especially in calendar making. Of course no other 
planet has a tropical year, so far as dwellers on the earth 
are concerned. But the other planets all have another 
important kind of year, which the earth does not have. 
It is called the Synodic year and corresponds to the synodic 
period (p. 161) in the case of the moon. To define it, sup- 
pose, in Fig. 55, we have the 
orbits of the earth and Jupiter. 
For both planets the sidereal 
year is the time required to 
complete revolutions from any 
two points such as E and J back 
again to the same points. But 
for Jupiter, which has a synodic 
year, this synodic year is defined 
J " as beginning when a straight 

Fig. 55. Synodic Year. . ,, , ,, 

line drawn from the earth to the 
sun at S passes through Jupiter at J. And, similarly, the 
synodic year ends when the revolutions of both bodies make 
it again possible to draw a straight line from the earth to 
Jupiter through the sun. 

We have here supposed the orbits of both earth and 
Jupiter to lie in a single plane. This may be done as a 
first approximation for all the planets, since none of their 
orbits lie in planes very greatly inclined to the ecliptic plane, 
in which the terrestrial orbit is situated. 

Both the sun and Jupiter are seen from the earth pro- 
jected on the background of the celestial sphere; conse- 

208 




THE PLANETS 

quently, when they are in this straight-line position, they 
should appear to us at the same point on the sky. Owing 
to the existing small angle between the orbit planes, it will 
happen only rarely that they will appear to occupy the same 
point quite exactly. So the synodic year is considered to 
commence when they are as nearly as possible in a straight- 
line position, and therefore in the closest possible apparent 
proximity, as seen by us projected on the sky. 

At such a time, we say that Jupiter is in Conjunction 
with the sun. In general, the term " conjunction " is thus 
used whenever two celestial bodies are in very close prox- 
imity, as seen from the earth, projected on the celestial 
sphere. 

A very simple mathematical relation exists between the 
synodic and sidereal periods of any planet. It is based on 
the fact that the synodic period depends on a line passing 
through the earth as well as the planet, and must therefore 
be affected by the terrestrial as well as the planetary rate 
of orbital motion ; while the sidereal period depends on the 
planetary motion alone. 1 

The foregoing reasoning applies strictly to those planets 
only whose distances from the sun are greater than that of 
the earth from the sun. These are called Superior planets 
to distinguish them from Mercury and Venus, which are 
accordingly called Inferior planets, because their orbits lie 
within that of the earth. 

These inferior planets, of course, have sidereal and synodic 
periods defined in the same way as the corresponding 
periods of the superior planets. The accompanying Fig. 56 
represents the case of an inferior planet such as Venus. 
The sidereal period of Venus, like that of Jupiter, is the 

1 Not. 27, Appendix, 
p 209 



ASTRONOMY 

time required by Venus to complete an orbital revolution 
around the sun, from any fixed star back to the same star 
again, supposed seen from the sun. But when we draw our 
straight line passing through the sun, the earth, and Venus, 
Fig. 56 shows that such a line can be drawn when Venus is 
in the position V, or in the position V. In either case, 
Venus and the sun will be seen from the earth close together, 
as projected on the celestial sphere; and will therefore be 
in conjunction. When Venus is thus in conjunction through 
being situated between the sun and the earth, we call the 
phenomenon Inferior Conjunction ; and 
when the sun is between Venus and the 
earth, we call it Superior Conjunction. 

Of course a superior planet, like Jupiter, 

whose orbit is entirely outside that of the 

earth, can never be placed between the 

FlG - p^ an e^ erior earth and the sun, and can therefore never 

have an inferior conjunction. Superior 

planets have superior conjunctions only; inferior planets 

have both inferior and superior conjunctions. 

The synodic period of Venus is, then, the time in days 
elapsing between two successive inferior conjunctions, or 
two successive superior conjunctions. But the mathe- 
matical relation connecting the synodic and sidereal periods 
is slightly different from that which holds in the case of a 
superior planet. 1 

The following little table contains approximate plane- 
tary periods ; and exhibits the interesting fact that both 
kinds of periods increase from Mercury to Mars, inclu- 
sive. Also, for this part of the table, the synodic periods 
are always the greater periods. But for all the other 

1 Note 28, Appendix. 
210 




THE PLANETS 




planets the synodic periods are far smaller than the si- 
dereal periods; and they are all nearly equal in dura- 
tion. 2 

It is plain that when any planet is in conjunction with 
the sun, we shall be unable to see it. Sun and planet being 
then projected on the sky at nearly the same point, the 
bright solar light will, of course, overcome the faint planet, 
and make it invisible. In other words, the planet, appearing 
near the sun, will be above the horizon in daytime. To 
make the planet visible, it must be far from the sun, as seen 
projected on the sky; i.e. there must have been consider- 
able synodic motion since the time of conjunction. Visi- 
bility from the earth depends on synodic motion, not actual 
motion in the orbit. 

It is customary to use the term " elongation" to designate 
a planet's angular distance from the sun, as we see it pro- 
jected on the sky. At the time of conjunction, the planet's 
elongation is very small ; it may even be zero. We have 
seen in Figs. 55 and 56 the state of affairs when a conjunc- 
tion with the sun occurs in the case of a superior and inferior 

1 These periods have been used on pp. 50 and 51. 

2 Note 29, Appendix. 

211 



ASTRONOMY 




Fig. 57. Superior Planet. Greatest 
Elongation. 



planet. As the synodic motion advances after conjunction, 
the planets increase their elongation from the sun. Figures 
57 and 58 show the maximum elongations the two kinds of 

planets can attain. For the 
superior planets, like J, repre- 
senting Jupiter (Fig. 57), the 
elongation may reach 180°. For 
the inferior planets, like V, repre- 
senting Venus (Fig. 58), there is 
a certain definite maximum angle 
of elongation SEV, which occurs 
when there is a right angle at 
V ; i>e. when there is a right angle 
between the directions of earth 
and sun, as seen from the planet. 
When the elongation is 180° in the case of a superior 
planet (Fig. 57), the sun is directly opposite the planet, as 
seen from the earth, projected on the sky. Thus, in the 
figure, the sun would be seen from the earth E projected 
toward the upper part of the page, and Jupiter directly 
opposite, projected toward the lower part 
of the page. The planet is then said to be 
in Opposition. The greatest possible elon- 
gations (Fig. 58) for the inferior planets 
Mercury and Venus, which can never be in 
opposition, are 47° l for Venus, and 28° for 
Mercury. These numbers may be verified by 
means of a simple mathematical calculation. 2 

Let us still remember that for the purpose of a first ap- 
proximation we may consider all the planetary orbits to lie 
in a single plane, the plane of the ecliptic. It follows that 

1 Cf. p. 51. 2 Note 30, Appendix. 

212 




Fig. 58. Inferior 
Planet. Greatest 
Elongation. 



THE PLANETS 

we must always see the planets projected on the sky near 
the great circle cut out by that plane, — the ecliptic circle, 
in which we also see the sun projected. Now since Mercury 
is thus always near the ecliptic circle, and always within 28° 
of the sun, it must appear to us to oscillate back and forth 
near the ecliptic circle, appearing now on one side of the 
sun, now on the other. This is also true of Venus, the 
other inferior planet, though here the arc of oscillation is 
much greater, as we have seen. When Mercury is at either 
extreme of its oscillation, it is in greatest elongation. When 
it is an eastern elongation, Mercury being east of the sun, 
the planet is visible for a short time after sunset. When 
it is a western elongation, the planet is west of the sun, and 
is visible a short time before sunrise. But owing to the 
apparent proximity of the sun, Mercury is always projected 
against the rather bright background of the sky near the 
point where the sun rises or sets at the horizon. Thus 
Mercury is not very easy to see. Venus, with its much 
greater possible elongation angle, is a very easy object to 
the unaided eye. 

In general, we thus find that the visibility of an inferior 
planet depends on the production of these maxima of elonga- 
tion by the synodic motion (cf. p. 50). 

In the case of a superior planet the state of affairs is very 
different. Visibility still depends on synodic motion ; as 
before, the planets cannot be seen near the time of conjunc- 
tion. But as their synodic motion advances, these planets 
do not approach a moderate maximum elongation, and 
appear to oscillate back and forth across the sun. For, as 
we have already seen, the superior planets have their oppo- 
sitions when their elongation from the sun is 180° ; then 
they are directly opposite the sun ; and are therefore observ- 

213 



ASTRONOMY 

able on the visible part of the celestial meridian at mid- 
night, when the sun is on the lower and invisible part 
(cf. p. 51). 

But, nevertheless, the superior planets do have certain 
oscillations in their apparent motions among the stars, as 
seen from the earth. These oscillations cause them to per- 
form at times so-called " retrograde" motions, traveling 
apparently among the stars from east to west instead of 
west to east, which is their usual direction of apparent 
motion. Sometimes, too, they have temporary "station- 
. i ary points," appearing immobile for a short 
^ / j time, like fixed stars. 

To understand this state of affairs, let 
^ us consider for a moment the orbits of the 
earth and a superior planet like Mars. The 
accompanying Fig. 59 shows these orbits, 
not drawn to scale, but again supposed to 
be in a single plane, and circular. Be- 
Fig. 59. Retrograde ginning at the time of opposition, Mars, 
earth, and sun are shown on the line MES. 
At the end of one month, Mars will be at M ' and the earth 
at E'. After three months, Mars will be at M " and the 
earth at E" . At these three dates, therefore, terrestrial 
observers will see Mars projected on the sky along the three 
successive directions EM, E'M', and E"M". Both planets 
have been constantly and uniformly moving in the direction 
of the curved arrows, yet from E' we see Mars along E'M' \ 
apparently retrograded back of the direction EM, or contrary 
to the direction of orbital motion for both planets. At 
E"M" , Mars has again begun to move forward in its ap- 
parent motion among the stars, and that forward motion 
will evidently become more rapid a little later. It is also 

214 



THE PLANETS 

clear that about the time the apparent motion changes from 
retrograde to direct, Mars will for a short time appear 
quite stationary among the stars. And it is further evident 
from the figure that the middle of the arc of apparent retro- 
gression must occur about the time of opposition, when the 
planet is nearest the earth. 

There is but one more peculiarity of importance in con- 
nection with this apparent motion of the planets as seen 
from the earth, and projected on the sky. It arises from 
the fact that the orbital planes do not coincide accurately 
with the ecliptic plane, and therefore the planets do not 
always appear to us on the sky projected accurately on the 
ecliptic circle. They have certain small apparent motions 
toward the ecliptic circle, and again away from it. It 
follows that a planet's arc of retrograde motion does not 
simply return along the same line over which it traveled 
in its direct motion, as would be the case if all planetary 
motions were accurately in the ecliptic plane. The actual 
retrograde apparent motions usually involve peculiar curves, 
both for the superior and inferior planets. 

We shall close this chapter with another reference to the 
Keplerian method of determining the planetary periods. 
The matter could not be explained fully until the synodic 
period had been made clear. By the aid of that period, 
astronomers of old possessed still another simple way of 
ascertaining the sidereal period by observation. They could 
observe the date when the planet was in opposition to the 
sun, when it comes to the meridian at midnight. Then 
the interval between two successive oppositions is the synodic 
period (p. 208) ; and from the synodic period they could 
calculate the sidereal period, which is the true period of 
orbital revolution, by means of a simple mathematical 

215 



ASTRONOMY 

equation. 1 This method cannot be used for the inferior 
planets, as they do not have oppositions. 

The accuracy of this measurement of period could be 
increased greatly by comparing two oppositions between 
which the planet had made many revolutions around the 
sun. Thus, by a comparison of two oppositions separated 
by five hundred synodic periods, the error of observation 
affecting the exact times of opposition would, of course, be 
divided by 500. This was actually possible in the case of 
the principal planets, by utilizing existing ancient records 
of opposition observations. 

Furthermore, it was necessary to compare distant oppo- 
sitions, to eliminate the effects of orbital flattening in the 
case of both the planet and the earth. For it is clear that 
successive synodic periods will not be accurately equal : 
they would be so, if the orbits were truly circular ; but from 
the average of a large number of successive revolutions this 
source of error is practically removed. 

1 Note 27, Appendix. 



216 



CHAPTER XII 

THE PLANETS ONE BY ONE 

It will now be of interest to consider separately the many 
details in which the planets differ amongst themselves ; and 
we shall begin with Mercury, the one nearest the sun. As 
we know, it always appears projected on the sky in the 
vicinity of the sun (p. 50) ; sometimes on one side of it, 
sometimes on the other. The ancients did not perceive 
that this planet, seen alternately on opposite sides of the 
sun, was a single body. They had two names for it, — 
Apollo and Mercury. 

The seasons on Mercury must present a rather curious 
problem. We have no means of ascertaining with any 
degree of certainty the angle between this planet's rotation 
axis and the plane of its orbit (p. 203). On the earth this 
angle is 66 J° ; and it is owing to the existence of such an 
angle that we have the regular terrestrial seasons (p. 120). 
Therefore we know very little about the seasons of Mercury, 
so far as they may be analogous to terrestrial seasons. But 
we know that the distance of this planet from the sun has 
so large a variation between perihelion and aphelion that 
a very variable quantity of solar heat must reach it at 
different times. There must exist a variability from this 
cause, great enough to make very appreciable temperature 
changes. The interaction of this with possible seasons of 
the terrestrial kind may give rise to hot summers and cold 
summers, etc., in different years. 

217 



ASTRONOMY 

In the telescope, Mercury exhibits phases like our moon, 
and due to the same cause. It has little or no atmosphere 
in all probability; and most astronomers can see but the 
faintest surface markings. Lowell, however, has published 
drawings of Mercury showing many geometric lines and 
angles ; and he thinks they change their apparent positions 
on the planet very slowly. This would indicate that there 
is no rapid axial rotation like the earth's. If the planet 
turned quickly on its axis, the rotation would soon carry 
some of the markings out of sight around the edge of the 
planet (p. 202). Possibly, therefore, Mercury, like the moon, 
rotates on its axis but once, while making an orbital circuit 
around the sun in 88 days. If this be so, there must be a 
very hot hemisphere, always facing the sun, and a very 
cold opposite hemisphere. But this would be modified 
somewhat by the very large librations (p. 171), which would 
result from Mercury having an unusually flattened orbit 
around the sun. 

The surface of Mercury is not very brilliant. It has been 
calculated that it reflects only 13 per cent of the solar light 
falling upon it. This percentage of light-reflection is called 
the planet's Albedo ; and Mercury has the lowest albedo in 
the solar system. 

The planet Venus, the other inferior planet, is also seen 
alternately on opposite sides of the sun, appearing as morning 
and evening star. But it attains a much greater angular 
distance from the sun than does Mercury, and is also more 
brilliant. It is at times the brightest of all the planets, 
and can even be seen by the unaided eye in full day- 
light near the occasions of its greatest elongation from the 
sun. 

The telescopic phases of Venus range all the way from a 

218 



THE PLANETS ONE BY ONE 




Fig. 60. Ptolemy's Theory of Venus. 



complete circle down to a narrow crescent. 1 According to old 
Ptolemy's theory (p. 189), we should never see the phase of 
Venus larger than the half- 
moon shape. For Ptolemy 
supposed Venus moving on 
a circle whose center was 
always near a line joining 
the earth and the sun. It 
is clear, from Fig. 60, that 
the angle at Venus be- 
tween the earth and the 
sun could never be as 
small as a right angle; 
and so Venus could never 

show a phase bigger than the half-moon, according to the 
accepted Aristotelian theory of phase phenomena (p. 163). 

This matter is most in- 
teresting ; the moment 
Galileo turned the first 
astronomic telescope upon 
Venus, about the year 
1610, and saw a phase 
larger than the half-moon, 
he had at once a strong 
proof that something was 
wrong with this particular 
detail of the sacrosanct 
Ptolemaic theory. To 
remove this difficulty, 
however, it would only have been necessary for Ptolemy to 
lengthen the radius of the Venus epicycle. 

1 Well shown in Plate 8, p. 225, a reproduced photograph. 
219 




Fig. 61. Greatest Luminosity of Venus. 



ASTRONOMY 

Venus gives a good example to demonstrate that a planet 
does not attain its greatest luminosity when nearest the 
earth, nor when exhibiting the largest possible phase. Fig- 
ure 61 illustrates this problem. The points S, V, E repre- 
sent positions of the sun, Venus, and the earth at inferior 
conjunction (p. 210). Venus has then no perceptible disk ; 
we see its dark side. As time goes on, the phase of Venus 
grows; and the light reflected toward us increases in pro- 
portion to the increasing area of the visible disk. But 
at the same time the distance from Venus to the earth 
is increasing ; and the intensity of the planet's light, as 
received by the earth, of course diminishes rapidly with the 
increase of our distance from Venus. Thus, in the figure, 
at the moment when Venus has reached the point V, the 
earth is at E'. The area or phase of the visible disk has 
grown from zero at V to the segment shown unshaded at V, 
but the distance between the two planets has increased from 
VE to V'E'. 

Thus we see that, beginning with inferior conjunction, 
the disk area grows much more rapidly than the distance; 
consequently, Venus grows more brilliant to our eyes. 
But, later on, this is reversed ; so there must be a certain 
point where Venus suddenly begins to substitute a decrease 
of visible brilliancy for the previous increase. This is the 
moment of maximum luminosity, as seen from the earth ; 
it is a nice problem, requiring the infinitesimal calculus for 
its solution, to determine this moment exactly. It will 
suffice here to say that it occurs about 36 days from inferior 
conjunction, when Venus has a phase like the crescent moon. 

Venus is believed to possess an atmosphere, for it has a 
very high albedo, or light-reflecting power, which indicates 
a reflecting surface containing clouds. Moreover, Venus is 

220 



THE PLANETS ONE BY ONE 

occasionally seen to pass between the earth and the sun, 
— a phenomenon called a Transit of Venus. When these 
transits are about to occur, and just as the planet is be- 
ginning to encroach upon the solar disk, as seen from the 
earth, a ring of light becomes visible around the part of 
Venus not yet projected upon the sun. This cannot be ex- 
plained otherwise than as a refraction or reflection of solar 
light by the planetary atmosphere. 

Certain ill-defined shadings have been seen at times on 
the planet's surface : Lowell goes so far as to give a map of 
Venus showing very clear geometrical structures of straight 
lines. These are of interest because of their bearing on 
Lowell's observations and theories as to the Mars " canals." 
From observations of the markings he concludes that 
Venus (as he also found in the case of Mercury) has a very 
slow axial rotation ; that it probably turns on its axis in 225 
days, which is also its sidereal period. If this be correct, 
Venus must always turn the same face toward the sun. 

The planet Mars, which we shall next consider, differs 
greatly from Mercury and Venus. Its orbit is exterior 
to that of the earth and varies quite considerably from an 
exact circular form, so that the planet's distance from the 
sun, and its distance from the earth, undergo very wide 
variations, corresponding to the planet's motion in its orbit. 
Furthermore, unlike Mercury and Venus, Mars has certain 
very well-defined and constantly visible surface markings. 
These have enabled astronomers to ascertain with precision 
the length of the Martian day, or period of axial rotation. It 
is found to be about 24| of our terrestrial hours, or nearly 
the same as the day of our earth. The diameter of Mars is 
about half that of the earth ; and the inclination of its axis 
to the plane of its orbit around the sun is 65°. Since the 

221 



ASTRONOMY 

corresponding angle of inclination in the case of the earth is 
66^°, it is clear that the Martian seasons will resemble closely 
those experienced by ourselves. Thus there are many points 
of resemblance between the two planets, Mars and the earth ; 
and therefore is Mars the best hunting ground for those who 
seek a planet with intelligent inhabitants. 

In the telescope, Mars shows no crescent phases like the 
moon or the inferior planets, because its orbit is outside 
that of the earth; and so the angle at Mars between the 
earth and the sun can never be as big as a right angle. Its 
atmosphere should be less dense than that of the earth; 
for the absence of clouds is indicated by our seeing con- 
stantly permanent markings on the planet's own surface, 
and by the observable fact that Mars has an unusually low 
albedo, or light-reflecting power. Moreover, owing to its 
small size and small mass, the attractive force of gravity on 
Mars is less than half that existing on the earth. Conse- 
quently, it is not improbable that Mars has been deprived 
of its atmosphere in the same way that the moon is believed 
to have lost its own air (p. 167). 

Mars has two satellites or moons ; and they are in some 
respects the most peculiar bodies in the solar system. Their 
special oddity arises from their close proximity to the planet, 
and the consequent shortness of their periods of orbital 
revolution about it. Deimos, the outer satellite, has an 
orbital period of 30 h 18 m . Phobos, the inner one, revolves 
in its orbit in 7 h 39 m . These brief intervals are the " lunar 
sidereal periods " (p. 161) for Mars. Now the planet itself 
takes about 24f hours to complete an axial rotation. There- 
fore the orbital motion of Phobos, as seen from Mars, makes 
it move among the stars from west to east much faster than 
the apparent diurnal motion of the Martian celestial sphere 

222 



THE PLANETS ONE BY ONE 

makes the satellite seem to move from east to west. In other 

words, Phobos rises in the west and sets in the east ! 

Deimos, however, with its period of 30 h 18 m , travels diur- 

nally from east to west. We can investigate easily its 

apparent diurnal motion. Its period being 30.3 b , in one 

360° 

hour it moves among the stars In the same time the 

6 30.3 

360° 
diurnal rotation of the Martian celestial sphere is — — • The 

24.7 

apparent motion of Deimos is therefore : 

360° 360° , , , . 

— — — — — - in an hour, east to west. 

It will therefore make an apparent rotation of 360° around 
the sky in a number of hours found by dividing 360°, the 
circumference of an entire diurnal circle, by the above 
hourly motion. The result of this division is 128 hours; 
and this is the " lunar day" (p. 176) of Deimos. And so we 
have the unusual condition that the lunar day is far longer 
than the lunar " month," or sidereal period. 

Approaching now the question of Martian " inhabitants," 
and their canals, we must first inquire as to the existence of 
water vapor in the atmosphere of the planet. Is there 
any? This is a matter of prime importance in connection 
with the famous supposed canals. If there exists on the 
planet a network of geometric markings, their explanation 
as waterways must stand or fall by the water vapor in the 
planetary air. A flow of water in canals can be imagined 
only if we suppose also evaporation of that water into an 
atmosphere, and subsequent precipitation of it as snow or 
rain. If this precipitation occurs for some reason principally 
near the planet's poles, while the evaporation takes place 
all along the canals, we might imagine the latter to have 

223 



ASTRONOMY 

been constructed artificially to carry the water away from 
the poles, so as to fructify and irrigate the entire planetary 
surface. 

When these markings were first seen by Schiaparelli the 
weight of observational evidence favored the presence of this 
water vapor. Such observational evidence is all obtained 
by means of an instrument called the spectroscope (p. 282). 
If the solar light reflected from Mars passes through a plane- 
tary atmosphere containing water vapor, its " spectrum/ ' 
as seen in the spectroscope, will show certain bands called 
water vapor bands. Unfortunately, we cannot observe 
the Martian light until it has passed through the terrestrial 
atmosphere, which always contains some water vapor. The 
difficulty is to determine whether any observed vapor bands 
are due to the Martian atmosphere, or to that of the earth. 

There is but one way to distinguish between the two : we 
must compare the Martian spectrum with that of our moon. 
The lunar spectrum will show no water vapor effects except 
such as are due to the earth's air, for the moon itself has no 
atmosphere. Consequently, a lunar observation gives us 
only terrestrial water vapor bands ; a Mars observation gives 
us the terrestrial plus the Martian effects. Any observable 
difference between the two is due to Martian vapor alone. 

It is clear that this method of observation cannot be suc- 
cessful if there is very much water vapor in our own air. If 
there is, the slight difference between the moon and Mars will 
be masked completely. As existing observations have been 
found discordant, Campbell made an expedition to the 
summit of Mt, Whitney (15,000 ft.) in 1909. He took the 
necessary instruments with him and photographed the 
spectra of Mars and our moon at the exact time when Mars 
was most favorably situated in proximity to the earth. 

224 






Photos by Barnard. 
PLATE 8. Mars and the Crescent Venus, 



THE PLANETS ONE BY ONE 

At the great elevation of Mt. Whitney there was but 
little terrestrial atmosphere above the observer. The lunar 
spectrum, which exhibits the effects caused by our own 
air, showed so little water vapor that Campbell concludes 
it would never have been detected at all by a person 
previously ignorant of its existence. And the Martian 
spectrum was equally destitute of water vapor bands, 
even the faintest. On account of this elimination of the 
earth's air, these observations must be considered by far 
the most reliable in our possession. They seem to settle 
the water vapor question in the negative ; with the Martian 
vapor the Martian water goes; and, without water, the 
canals are impossible as artificial waterways. 

The markings on Mars of which we are certain consist of 
various permanent patches, lines, and areas of different 
shades; and there are also two bright spots at the poles 
known certainly since the time of Herschel. The accompany- 
ing Plate 8 contains a series of photographs of Mars, showing 
these markings and white spots very plainly. There is also 
distinct evidence of axial rotation, the exposures having been 
made in sets of three, with an interval of l h 22 m between the 
first and last set. In that interval the markings have 
moved perceptibly across the disk (p. 202). The lower 
part of the plate shows a photograph of Venus, in the 
crescent phase (p. 219). The polar spots seem to increase 
in the Martian winter season, and to diminish in the summer. 
If so, they may be ice-caps ; and it is this notion that gives 
color to the canal theory. For the melting ice-caps, in 
this theory, are used as the source of water to be pumped 
through the canals. Later, the water evaporated out of 
the canals is supposed to be returned to the poles by atmos- 
pheric movements, and there again precipitated as snow. 

q 225 



ASTRONOMY 

But if the planet, in comparison with the earth, is as cold as 
it should be according to its distance from the sun, there is 
quite a possibility that the caps are not ice at all, but perhaps 
some other substance, such as solidified carbon dioxide. 

It is not difficult to calculate the theoretic temperature on 
the surface of Mars, and it is found to be —33° on the 
Fahrenheit scale. 1 Since this low theoretic temperature 
would negative the existence of water in an uncongealed 
state, the advocates of canals are compelled to assume 
a heavy blanket of atmosphere, charged with much water 
vapor, to keep Mars warm, as it were, and cause the actual 
surface temperature to be far above its theoretic value. 
Such an atmosphere might act in this way ; but here again 
we find the absolute necessity of assuming the presence of 
water vapor in spite of observational evidence to the con- 
trary; and we are asked to imagine Mars to be an arid 
desert requiring irrigation, and yet above this arid desert 
a wet, foggy atmosphere, highly charged with water vapor. 

In view of the great public interest in this Mars matter, 
we shall venture to quote briefly from an article published by 
the writer a few years ago. First, as to the question : Do 
the geometric markings really exist ? The evidence here is 
almost all positive. Most astronomers who have observed 
Mars under favorable conditions and with powerful telescopes 
have seen markings, but the number of lines reported varies 
from several hundreds down to two or three. Finally, a 
very few prominent markings have been photographed. 

Let us first consider the visual evidence. Let us examine 
the witnesses, for that is what these astronomers really are, 
eye-witnesses of the lines on Mars. Some years ago, 
Lowell observed on the planets Venus and Mercury certain 

1 Note 31, Appendix. 
226 



THE PLANETS ONE BY ONE 

systems of geometric markings. As it is impossible to 
suppose that all planets possess intelligent engineers, it is 
essential to the Martian theory to show that these Venus 
markings are quite unlike those now seen on Mars. Ac- 
cordingly, in his book entitled Mars and its Canals, pub- 
lished in 1906, Lowell refers to these older Venus observa- 
tions in the following words : "The Venusian lines are hazy, 
ill-defined, and non-uniform." But in the original article 
in which he described what he saw on Venus, 1 we find the 
following : "They (the markings on Venus) are not shadings 
more or less definite, but perfectly distinct markings. I 
have seen them when their contours had the look of a steel 
engraving." 

The only way in which these two statements concerning 
Venus can be brought into accord is to suppose that in the 
interval of nine years the observer has, for some reason, 
changed his opinion. 

Additional important testimony is furnished by Mr. A. E. 
Douglass, who was chief assistant at the Lowell observatory 
in Flagstaff for seven years, from 1894 to 1901, and since then 
held the position of astronomer there for a considerable 
period. In May, 1907, he published an article in the Popular 
Science Monthly, entitled "Illusions of Vision, and the 
Canals of Mars." This title alone shows that he had 
changed his views ; and his actual words in the 1907 article 
are : 

"The ray illusion (sic) is to me a very satisfactory explana- 
tion of many faint canals . . . the only objective reality 
is the spot from which they start." 

Again, speaking of what he calls the "halo illusion," he 
says : 

1 Monthly Notices of the Royal Astronomical Society, London, 1897. 

227 



ASTRONOMY 

"The double canals of Schiaparelli in 1881-2 and of 
Perrotin and Thallon in 1886 . . . are . . . due to this 
cause.' ' 

And again : 

"Thus, in conclusion, we see that there are fundamental 
defects in the human eye producing faint canal illusions/ ' 

Having thus outlined briefly the apparent contradiction in 
LowelPs testimony and the reversal of Douglass', it will 
be of interest to explain how it may be possible for observers 
to be in error to such an extent. For this purpose, we 
must mention some of the possible causes that may impair 
the correctness of an observation. At least five im- 
perfections come into play: imperfections of the earth's 
atmosphere, the telescope, the eye, the optic nerve, and 
the imagination. The process of seeing a thing is not 
at all simple. Light waves coming from the object under 
examination, after passing through the atmosphere and 
telescope, fall upon the outer surface of the eye. They are 
concentrated, or focused, by the lens in the eye, and pro- 
duce an effect, which we do not quite understand, upon 
the retina at the back of the eye. This, in some un- 
explained way, results in an impression being received 
by the brain through the optic nerve. Then, the brain 
in its turn does an unexplained something with that im- 
pression ; what we think we see is equal to that which 
came through the eye and optic nerve, plus what the 
brain does with it later. The mind cannot distinguish 
between an impression caused by the eye and optic nerve 
and one produced by action of the brain itself. 

Now it is important to remember that imperfections of 
the atmosphere, such as clouds, and all imperfections of 
the telescope, generally tend to diminish or destroy the 

228 



THE PLANETS ONE BY ONE 

possibility of vision ; but those of the eye and imagination, 
if they act, are just as likely to increase the number of 
details we think we see. Especially when the object is faint 
and indistinct, — trembling, as it were, on the very limit 
of visibility, — then especially can a very slight activity 
of the imagination either prevent our seeing it, or bring 
it seemingly into view. And this extreme faintness ad- 
mittedly exists in the case of almost all the Martian mark- 
ings. 

This theory explains why highly experienced observers 
see so much more than beginners. They think they are 
training the eye, so as to increase its powers, while in reality 
they may only be training that slight imperfection of the 
imagination which tends to increase details thought to be 
visible. The theory also furnishes an explanation of the 
fact that so considerable a number of observers think they 
have seen the faint canals. Nothing more strongly increases 
the powers of imaginary seeing — of seeing the unseen — 
than the knowledge that others have already made the ob- 
servation. We are very prone to "see" what we are told 
by others is visible : we think we see what we desire and hope 
to see ; do what we will, we cannot prevent this. 

Coming now to a consideration of the photographic ob- 
servations, we must mention one or two matters that are 
not well known to the general public. In the first place 
the size of a Mars picture made by direct exposure of a 
photographic plate at the focus of the Lowell telescopes is 
not larger than the head of an ordinary pin. From so small 
a picture we could not even hope to discover any details. 
Therefore we must enlarge it as much as possible ; and there 
are two ways of doing this. The first is to place an enlarging 
lens in the telescope itself. Two disadvantages limit this 

229 



ASTRONOMY 

method. First, it complicates the optical system of the 
telescope, with consequent loss of distinctness in the image ; 
and secondly, it makes the image on the plate less brilliant. 
The cause of this loss of brilliancy is simple. The total 
quantity of light received from the planet is constant ; if, 
therefore, we spread it over an enlarged surface, each part of 
that surface will receive less light. For instance, with an 
enlargement of five diameters, the surface of the image is 
25 times as large. The resulting diminution of light makes 
necessary a longer exposure of the photograph, and a con- 
sequent increased difficulty in making the clock mechanism 
attached to the telescope follow with exactness the motion of 
Mars in the sky. Experiment has shown the greatest photo- 
graphic enlargements that can be made in this way with 
the Lowell telescopes ; and the negatives of Mars, including 
the most recent ones made in the Andes, never exceed three- 
sixteenths of an inch in diameter. 

The other method of increasing the size of photographs 
is to use an ordinary enlarging camera, after the telescopic 
negative has been finished. There is here no difficulty in 
securing sufficient light, as in the case of enlargements made 
in the telescope itself. For we can use artificial illumination 
of the original negative, and make this illumination as 
strong as may be necessary. But there is another serious 
difficulty. Every photographic negative is developed by 
placing the plate in a chemical bath, after it. has been ex- 
posed to light. This results in the precipitation of silver 
particles upon the plate, wherever its sensitive surface has 
been exposed to light. The picture is thus built up of sep- 
arate particles of silver. These particles are so small 
that the eye cannot distinguish the separate ones ; they 
run together, as it were, to form the picture. But the case 

230 



THE PLANETS ONE BY ONE 

is very different if we magnify the negative. We then see 
the separate grains of silver,- scattered here and there about 
the surface, and the picture itself is lost altogether. The 
same difficulty occurs if we attempt to examine any photo- 
graph with an ordinary microscope of considerable power. 
The separate silver grains at once appear, and the picture 
effect is lost. 

All this photographic experimentation, therefore, has not 
yet resulted in good pictures more than three-sixteenths 
of an inch in diameter and produced by purely photo- 
graphic processes, though somewhat larger negatives may 
possibly be made in the future. 1 All larger published 
pictures have been reproduced from hand drawings, and 
are therefore simply visual observations. The alleged 
photographic verifications have been made by the same 
observers who have studied Mars in the visual telescope ; 
again the eye, optic nerve, and brain were brought into play, 
and exactly the same causes as before impair the reliability 
of these visual observations of photographs. 

We conclude that neither by visual nor by photographic 
evidence has the existence of an artificial network of markings 
been proven, or even rendered highly probable. Therefore 
the time has not yet come when we shall have to inquire 
whether geometric lines indicate the presence of intelligent 
inhabitants ; that time will arrive if the lines themselves are 
ever shown to possess a real or even a highly probable 
existence. 

We shall next consider the Planetoids, or Minor Planets 
(pp. 183, 196). A large number of these tiny bodies travel in 

1 Those shown in Plate 8, p. 225, made with the 40-inch Yerkes' tele- 
scope, are about twice as large as the Lowell photographs ; and they show 
no signs of geometric canal networks. 

231 



ASTRONOMY 

orbits situated between Mars and Jupiter ; up to the present 
time several hundreds have been discovered and their orbits 
and motions computed. 

Much interest attaches to the history of the first one 
ever observed, — Ceres. It had long been noted that the 
space between Mars and Jupiter was an exceptionally 
large empty space in the solar system ; and it seemed strange 
that no planet should exist there. The matter appeared 
still more peculiar after Bode's empirical law was published 
(p. 196) in the latter part of the eighteenth century ; for this 
law indicated that there should be a planet between Mars 
and Jupiter. And in 1781 this indication received stronger 
confirmation, when the older Herschel found Uranus, one 
of the modern exterior planets, in or near the position 
predicted by the law. An astronomical society was accord- 
ingly organized to make a systematic search for the expected 
unknown planet. But not until the first day of the nine- 
teenth century did the long-sought object reveal itself; 
and to an independent observer in the Italian city of Palermo. 
There Piazzi was making an accurate catalogue of the fixed 
stars. Every night he made telescopic observations from 
which he could compute stellar right-ascensions and dec- 
linations (p. 34) ; and he planned to enter in a catalogue 
these two coordinates for every star in the sky, bright enough 
to come within the range of his telescopic power. 

But he did not confine himself to single observations. 
Each night's work was checked by careful repetition on 
several other nights. Sometimes he found an error, which 
usually consisted in the discovery of a star that had escaped 
his notice at some previous observation. But on this historic 
occasion, he found that a star was absent, although he had 
observed it on another night. And, strange to say, there 

232 



THE PLANETS ONE BY ONE 

was also an additional star close by, one that had apparently 
remained unobserved on the previous night. The conclusion 
was irresistible that the new star was the same one he had 
observed before, and that it must have moved among the 
other stars in the interval. This motion among the stars 
(p. 10) is the distinguishing characteristic of planets. A 
third observation made the matter sure : the second star 
was again absent, and a third new star once more appeared 
in a place previously vacant. The apparent motion between 
the second and third observations was proportional, both in 
magnitude and direction, to that between the first and second 
observations. So it must surely be a wandering star, — a 
true planet. Discovery and fame were his. 

But Piazzi was able to observe the new planet during a 
few weeks only, on account of illness. When news reached 
the astronomers of northern Europe, Ceres had already 
passed so near conjunction (p. 209) with the sun that further 
observations were impossible. There was well-grounded 
fear that the planet would not be found again ; for astrono- 
mers at that time had no good method of determining a 
planet's orbit from observations extending through such a 
short time. The older planets had been observed through 
many complete revolutions, and there was never any danger 
of their being lost, because they are bright enough to be 
visible easily by the unaided eye. 

But there was a young astronomer at Gottingen, Gauss by 
name, who succeeded in solving this difficult problem; 
and from his published orbit and ephemeris it was easy 
to find the planet again as soon as the apparent motion of 
the sun in the ecliptic had brought the planet to a position 
where it could again be sought in darkness. 

A year later, in 1802, the second minor planet Pallas was 

233 



ASTRONOMY 

found. In 1804 Juno was added; and in 1807, Vesta. It 
was not until 1845 that another appeared ; and three more 
in 1847. From that time on discovery proceeded but 
slowly, because the method in use was still the tiresome and 
arduous process employed by Piazzi. But in 1891, Wolf 
attacked the problem photographically, the photographic 
method having just commenced to be widely applied to 
astronomic purposes. His procedure was perfectly simple. 
A photographic plate was exposed in the telescope for 
several hours ; and care was taken to make sure that clock- 
work attached to the telescope moved it accurately during 
those hours, so as to keep pace exactly with the diurnal 
rotation of the celestial sphere. 

The photograph, when developed, would of course show a 
round dot corresponding to each fixed star within the field 
of view of the telescope. But if there was a wandering planet 
in range, it would move slightly with respect to the stars 
during the period of photographic exposure; and conse- 
quently its image in the picture would be drawn out into a 
short line, instead of appearing as a round dot like the 
stellar images. Thus the presence of a line would infallibly 
betray the existence of a planet. As many as seven plane- 
toids have been thus found on a single plate ; so the method 
is enormously effective. To it we owe an immense increase in 
our minor planet knowledge during the past twenty years. 

Plate 9 is a photographed field of stars, with two planetoid 
lines, or " trails." They will be found near the middle of 
the picture, as indicated by the marginal arrows. The 
trails are not quite parallel, showing that the orbit planes of 
these two planetoids are inclined at slightly different angles to 
the ecliptic plane. The difference in thickness of the trails 
indicates a difference of luminosity in the two planetoids. 

234 



> 




t 



Photo by Barnard. 



PLATE 9. Discovery of Planetoids. 



THE PLANETS ONE BY ONE 

The orbits of the small planets present some interesting 
peculiarities. There are several open spaces where practi- 
cally no orbits appear. Curiously enough, these open 
spaces occur at points where the minor planet periods of 
orbital revolution in accordance with Kepler's harmonic law 
(p. 188) bear a simple relation to the period of Jupiter. 
It was long ago explained by Lagrange that if two planets 
have periods connected by a simple proportion, such as f , j, 
|, etc., then persistent perturbations (p. 206) must result, 
which will gradually change the orbits until the simple 
relation is destroyed. It is in accord with this principle 
that Jupiter has forced the minor planet orbits out of these 
critical positions in space, and made them congregate at 
intermediate positions. 

As to the size of the planetoids, it has been computed 
that the mass of the entire group can be but a small fraction 
of the earth's mass. The individual planetoids are probably 
not more than one ten-thousandth as massive as the earth, 
and their diameter will not average more than twenty miles. 

As to the evolution of these minor planets, there is not much 
doubt. If we accept the hypothesis of Laplace, usually 
known as the Nebular Hypothesis, the planets were formed 
by the concentration of matter thrown off from the sun in 
early ages, while it was still in a gaseous or nebulous con- 
dition. This matter is supposed to have been detached 
from the central mass in the form of a ring : we have only 
to imagine the minor planets an exceptional case, in which 
the ring, after breaking up, was prevented from concentrating 
into a single body by the perturbative action of the big planet 
Jupiter. Any other hypothesis as to the early history of 
the planets must, of course, also explain the planetoids as 
an exceptional case in cosmic evolution. 

235 



ASTRONOMY 

Among the minor planets is one very remarkable one, 
discovered by Witt in 1898 and by him named Eros. Its 
orbit comes well within that of Mars, and it approaches the 
earth at times nearer than any other planetary body except 
our own moon. It can pass within about 13J million miles 
of the earth ; and this makes it an especially valuable planet to 
observe for the purpose of ascertaining by certain indirect 
methods the distance from the earth to the sun. It is 
altogether probable that observations of Eros will give us 
ultimately the most accurate value of the sun's distance 
yet attained. There will be a very favorable opportunity 
to attempt the necessary observations in 1931 (cf. p. 263). 

Proceeding outward from the sun, we now reach the 
planet Jupiter, the largest in the solar system, and the most 
brilliant object in the sky at night, with the possible occa- 
sional exception of Venus. In the telescope Jupiter is a 
magnificent object, second only to Saturn in interest. It 
surpasses Saturn in size, but it lacks the splendid, calm 
mysterious ring. Markings of a more or less permanent 
character exist ; they look like cloud-belts running along the 
planet's equator. And clouds they doubtless are ; for Jupiter 
must have retained, and must possess, a deep layer of atmos- 
phere, on account of his very high gravitational attraction 
(see pp. 167, 222) ; and since there is also a high albedo, or 
reflecting power, we should expect the outer surface to be 
made of clouds, which have this power in a high degree. 

Jupiter's rotation period, or day, can of course be deter- 
mined from the markings. It is 9 11 55 m ; but there is some 
uncertainty in this period, because the cloud-markings 
probably have a drift of their own on the planet's surface, 
and thus do not determine the rotation with precision. 
The axis is only 3° out of perpendicularity with the orbit 

236 



THE PLANETS ONE BY ONE 

plane ; so that there should be no considerable seasonal 
differences of temperature. But the average Jovian tempera- 
ture must be very high, for the constant visibility of clouds 
indicates a hot surface temperature. If this be correct, 
Jupiter must be in a condition slightly resembling that of 
the sun. It must furnish its own heat, for it is too far 
from the sun to receive much thermal assistance from that 
body. 

Jupiter has eight moons, of which four can be seen with 
a small telescope. They are very interesting historically, 
for their discovery in 1610 by Galileo gave its death-blow 
to the old Ptolemaic theory of the universe. The follow- 
ing is a brief account of this great discovery, partly quoted 
from an article by the writer, first printed in the New York 
Evening Post. 

What must have been Galileo's feelings when he first 
found with his "new" telescope the satellites of Jupiter? 
They were seen on the night of Jan. 7, 1610. He had al- 
ready viewed the planet through his earlier and less powerful 
glass, and was aware that it possessed a round disk like the 
moon, only smaller. Now he saw also three objects that he 
took to be little stars near the planet. But on the following 
night, the three small stars had changed their positions, and 
were now all situated to the west of Jupiter, whereas on the 
previous night two had been on the eastern side. He could 
not explain this phenomenon, but he recognized that there 
was something peculiar at work. Long afterwards, in one 
of his later works, translated into quaint old English by 
Salusbury, he declared that "one sole experiment sufficeth 
to batter to the ground a thousand probable Arguments." 

The 9th was cloudy, but on the 10th he again saw his 
little stars, their number now reduced to two. He guessed 

237 



ASTRONOMY 

that the third was behind the planet's disk. The positions 
of the two visible ones were altogether different from either 
of the previous observations. On the 11th he became sure 
that what he saw was really a series of satellites accom- 
panying Jupiter on his journey through space, and at the 
same time revolving around him. In the 12th, at 3 a.m., he 
actually saw one of the small objects emerge from behind 
the planet ; on the 13th he finally saw four satellites. Two 
hundred and eighty-two years were destined to pass away 
before any human eye should see a fifth. It was Barnard, in 
1892, who followed Galileo. 

To understand the effect of this discovery upon Galileo 
requires a person who has himself watched the stars, not as 
a dilettante, seeking recreation or amusement, but with 
that deep reverence that comes only to him who feels — nay, 
knows — that in the moment of observation just passed 
he, too, has added his mite to the great fund of human knowl- 
edge. Galileo knew, on that 11th of January, 1610, that 
the memory of him would never fade ; that the very music 
of the spheres would thenceforward be attuned to a truer 
note, if any would but hearken to the Jovian harmony. 
For he recognized that the visible revolution of these moons 
around Jupiter, while that planet was itself visibly traveling 
through space, must end the old Ptolemaic theory of the 
universe. Here was a great planet, the center of a system 
of satellites, and yet not the center of the universe. Surely, 
then, the earth, too, might be a mere planet like Jupiter, and 
not the supposed motionless center of all things. 

The most interesting phenomena about the Jovian satel- 
lites are their frequent eclipses and transits (p. 13). Any 
satellite may be eclipsed to us, either through passing 
behind the ball of the planet, or by moving into such a 

238 



THE PLANETS ONE BY ONE 

position that the planet interposes between the satellite 
and the sun. In the latter case, the satellite receives no 
sunlight to reflect in our direction, and so becomes invisible. 

At other times a satellite will " transit" between us and 
Jupiter. Then it generally becomes invisible, too, unless 
it happens to be projected against a dark part of the Jovian 
surface, such as one of the cloud-belts. Finally, a satellite 
may pass between Jupiter and the sun, when its shadow is 
thrown on the planet's surface, and is plainly visible as a 
round black dot slowly crossing the bright planetary disk. 
None of the transits or eclipses occur suddenly : the satellites 
all have disks of sensible magnitude, and thus encroach upon 
the planet's edge very gradually. 

Observations of the exact time of these satellite eclipses 
are useful as an easy approximate method of determining 
terrestrial longitudes. If we note the instant of local time 
when an eclipse occurs, and compare it with the calculated 
Greenwich time, as given in the astronomical almanac, we 
ascertain at once the time difference of the observer's 
position on the earth, measured from Greenwich. And this, 
multiplied by 15, gives his longitude at once in degrees 
(p. 74). This method requires no instruments beyond an 
ordinary small telescope ; but it is not very precise on ac- 
count of the impossibility of observing the exact instant 
when the eclipse happens. Somewhat higher precision, 
with equal simplicity in method, may be secured from ob- 
servations of star occultations (p. 166). 

It was from observations of Jupiter's satellite eclipses that 
Roemer, in 1675, first ascertained that light is not propagated 
through space instantaneously, but requires an appreciable 
time for its transmission. He used a long series of satellite 
eclipses, and found they did not succeed each other at equal 

239 



ASTRONOMY 



intervals. During half the year they came too soon, by 
gradually increasing amounts; during the other half-year 
they came too late, by similar quantities of time. Roemer 
soon found that they came too soon when the earth was 
approaching Jupiter ; too late when it was increasing its dis- 
tance from Jupiter. He concluded correctly that they came 
too soon because the earth's approach to Jupiter diminished 
the distance the light traveled before reaching the earth, after 
the eclipse actually occurred. Consequently, the light 

arrived at the earth 
sooner, and observers 
saw the event sooner, 
too. It is clear from 
Fig. 62 that the extreme 
difference between ac- 
celerated and retarded 
eclipses must be the 
quantity of time required 
by light to cross a diame- 
ter E±E 2 of the earth's or- 
bit. This is found, by ob- 
servation of the eclipses, 
to be 998 seconds. It fol- 
lows that when the earth is at E h an eclipse will be observed 
998 seconds sooner than it would have been observed if the 
earth had remained at E 2 instead of traveling around its 
orbit. Bradley knew of Roemer' s observations when he 
explained aberration (p. 136), nearly a century later, as a 
result depending on the velocity of light and the earth's 
velocity in its orbit. 

Since modern laboratory experiments have made known 
that light moves at the rate of 186,000 miles per second, we 

240 




Fig. 62. Roemer's Discovery. 



THE PLANETS ONE BY ONE 

have only to multiply this number by 998 to obtain the diam- 
eter of the earth's orbit in miles. This gives 186,000,000 
miles, approximately ; and this number is correct. 

The planet Saturn is the last of the large planets observed 
by the ancients. The most interesting thing about it is its 
magnificent ring system, — a series of three disk-like rings, 
situated nearly in the plane of the planet's equator. The 
history of their discovery is worth noting. We hear of them 
first from Galileo, who saw a couple of " handles" or ansce 
attached to the planet in 1610. He was unable to explain 
them ; and, when he looked for them again on a later date, 
was unable to see them at all. The story is that he gave 
them up as inexplicable. 

Nearly half a century later, in 1656, Huygens published a 
book De Saturni Luna Observatio Nova, in which he announced 
the discovery of a satellite, and also gave a correct explana- 
tion of the mysterious ansce. But Huygens was not quite 
certain that his explanation was right. He was most 
anxious to secure for himself the priority of discovery, and 
yet he was unwilling to make a premature and possibly in- 
correct announcement. So he resorted to the ingenious 
device of a "logogriph," or puzzle. It appears in the 
De Saturni Luna as follows : 1 

aaaaaaa ccccc d eeeee g h iiiiiii 
1111 mm nnnnnnnnn oooo pp q rr s 
ttttt uuuuu 

It was not until 1659, three years later, in a book entitled 
Sy sterna Saturnium, that Huygens rearranged the above 
letters in their proper order, reading : 2 

1 It may be found in 's Gravesande's edition of Huygens, Lugduni Ba- 
tavorum, 1751, p. 526. 

2 Same edition, p. 566. 

r 241 



ASTRONOMY 

" Annulo cingitur, tenui piano, nusquam cohaerente, ad 
eclipticam inclinato. ' ' 

At the same time, 1 he re-published a series of drawings 
exhibiting several incorrect interpretations of the ring 
phenomena, as observed by various older astronomers. 
These are reproduced in Plate 10 : Fig. 1 is by Galileo, ob- 
served 1610; Fig. 2, by Scheiner, 1614; Figs. 3, 8, 9, 13, 
by Ricciolus, 1640-1650; Figs. 4, 5, 6, 7 by Hevelius; 
Fig. 10, by "EustachiusdeDivinis," 1646-1648; Fig. 11, by 
Fontana, 1646 ; Fig. 12, by Gassendi and Blancanus. Under 
these reproductions from Huygens we have placed a fine 
drawing made by Barnard with the Yerkes 40-inch telescope, 
Dec. 12, 1907. 

It will be observed that by the publication of the logo- 
griph of 1656, Huygens secured for himself the credit of 
what he had done. If any other astronomer had pub- 
lished the true explanation after 1656, Huygens could have 
proved his claim to priority by re-arranging the letters of his 
puzzle. On the other hand, if further researches showed that 
his explanation was wrong, he would never have made 
known the true meaning of his logogriph, and would thus 
have escaped the ignominy due to publishing an erroneous 
explanation. So the method of announcement was com- 
parable in ingenuity with the Huygenian explanation itself. 

The ring phases admit of easy explanation. The rotation 
axes of all revolving bodies maintain constant directions in 
space, unless disturbed by attractions such as cause the 
earth's axis to produce the precession of the equinoxes (p. 
129). Therefore the plane of the rotating rings must like- 
wise always maintain an unvarying direction in space. Now 
if this plane of the rings is imagined extended outward, 

1 Same edition, p. 634. 
242 




PLATE 10. Saturn. 



THE PLANETS ONE BY ONE 



until it cuts the celestial sphere, it will trace out a great 
circle there. This circle necessarily meets the ecliptic 
circle in two opposite points (cf. Fig. 6, p. 35), which are 
called nodes ; and it so happens that the angle between 
the great ring circle and the ecliptic is 28° on the celestial 
sphere. 

Saturn revolves in its orbit around the sun in a period of 
about 30 years. Therefore, it must pass one of the nodes 
every fifteen years, approximately. When Saturn is thus 




Fig. 63. Phases of Saturn's Ring. 

projected at one of the nodes, the sun, in its apparent 
motion along the ecliptic, may happen to appear in the 
other node at the same time. These positions are illustrated 
in Fig. 63 ; but in using this figure, it must not be forgotten 
that Saturn's orbit around the sun is very nearly in the 
ecliptic plane, in which the earth's orbit is also located. Let 
the sun, then, be at S, and the earth at E. Thus the sun ap- 
pears projected on the celestial sphere at the node S'. 
Saturn, located at H in its orbit, is projected at the same time 
in the other node at H\ It is evident that the earth must 
then lie on the line HS', the intersection of the two planes, 

243 



ASTRONOMY 

so that it is temporarily in the plane of the ring as well as 
in the ecliptic plane. 

When the earth is thus in the ring plane, we must see the 
ring edgewise ; and it is so thin that it then becomes quite 
invisible, except to the most powerful modern telescopes. 
It disappears, as Galileo found. Furthermore, near these 
times of disappearance, the earth may be for a short time 
on either side of the ring plane. And unless Saturn is quite 
accurately at the node, the sun will also be a little on one side 
of the ring plane or on the other. But the ring is illuminated 
on one side only, — the side toward the sun. Consequently, if 
the sun happens to be on one side of the ring plane while the 
earth is on the other, we observe the dark side of the ring- 
system. It should then also be invisible ; but powerful 
telescopes will still show it, appearing like a fine line of 
light. This is well seen in Barnard's drawing (Plate 10, 
p. 242), together with certain condensations, or thick places 
in the ring. This drawing was made with the ring in the 
edgewise phase : in Plate 11, we have added a fine photo- 
graph, also by Barnard, showing the open phase. This 
negative was made Nov. 19, 1911, with the 60-inch reflecting 
telescope at the Mt. Wilson observatory in California. 

As we have found, these times of ring-disappearance occur 
about once every fifteen years. In years near the periods 
of disappearance, the ring is seen nearly edgewise : it then 
looks like a very narrow oval or ellipse ; and it opens out 
to the widest extent about seven or eight years on either 
side of the date of disappearance. But the ring can never 
open into a circle, for the earth can never be elevated more 
than 28° above the plane of the ring, since 28° is the angle 
between the ring plane and the ecliptic plane, in which the 
earth is always situated. And the earth would need to be 

244 



THE PLANETS ONE BY ONE 

elevated 90° above the ring plane to enable us to see the 
ring as a circle. 

As to the constitution of the rings, we have very certain 
and most interesting knowledge. They are neither solid, 
liquid, nor gaseous, but consist of a dense swarm of tiny 
satellites, moving in orbits closely interwoven, and all 
lying in, or near, the plane of Saturn's equator. They are 
so numerous, and their orbits so closely packed and in- 
tertwined, that we cannot see between them, and so they 
look like a solid disk. They are not very unlike the group 
of planetoids, which are known to encircle the sun be- 
tween the orbits of Mars and Jupiter (p. 231). In 1857, 
Clerk-Maxwell proved mathematically that it is impossible 
for a system of solid or liquid rings to exist permanently. 
They would be in unstable equilibrium, and must infallibly 
break into a series of satellites. And this mathematical 
demonstration was abundantly verified observationally in 
1896, by Keeler. He observed the ring on both sides of the 
planet with the spectroscope. With this instrument, to be 
described later, it is possible to measure the linear velocity 
with which the edges of the ring approach the earth, or re- 
cede from it, as the ring performs its axial rotation around 
the polar axis of Saturn. Now it can be shown mathemati- 
cally that if the ring is really a mass of satellites, its outside 
edge should rotate more slowly than its inside edge. 1 On the 
other hand, if the ring is solid, of course the outside must move 
faster than the inside. Keeler found by actual measurement 
that the outside of the ring was moving 10 miles per second ; 
the inside, 12 J miles ; and he thus verified observationally 
the correctness of Clerk-Maxwell's mathematical conclusion. 

This observation of Keeler's is destined to rank as a 

1 Note 32, Appendix. 
245 



ASTRONOMY 

classic observation. We are given to regard astronomy as 
an ancient science, long since perfected, and incapable of 
further progress of importance. But this analysis of the 
ring constitution by methods purely mathematical; this 
theoretic prediction of invisible relative velocities of rotation ; 
and, finally, the complete observational verification by a 
method essentially novel, — all this constitutes a chain of 
scientific research worthy of standing at the side of the 
master work of the seventeenth century. 

Saturn has ten moons, in addition to the swarm composing 
the ring system. The largest (discovered by Huygens) is 
visible in small telescopes. Five were found before 1700 ; 
Herschel found two in 1789 ; Bond one in 1848 ; the other 
two were discovered photographically at Harvard College 
observatory within recent years. 

The next planet, Uranus, was discovered by Sir William 
Herschel in 1781. The history of Herschel, and of this dis- 
covery, is not without interest. He was the son of a Ger- 
man musician, was born in 1738, and came to England in 1757 
to seek his fortune. He settled at Bath, where he supported 
himself successfully as a music teacher. Although he worked 
very hard at his music, he found time to study also his 
favorite sciences of mathematics and astronomy. Having 
no instrument, he decided to make one ; but it was not until 
1774 that he succeeded in constructing a tolerable reflecting 
telescope. He wrote in 1783: "I determined to accept 
nothing on faith, but to see with my own eyes what others 
had seen before me." Four times he made a new telescope, 
each of greater size than the last ; and with each he made a 
re-survey of the entire visible heavens. On his tomb is 
graven the epitaph : 

" Ccelorum perrupit claustra." 
246 



THE PLANETS ONE BY ONE 

It was in the second of his celestial reviews, made with 
an instrument only seven feet long, that, as he says, "in 
examining the small stars in Gemini, I perceived one 
that appeared visibly larger than the rest. I suspected 
it to be a comet." Within a short time, Lexell was able 
to show that the motions of the new object could not 
be explained by any cometary orbit, and that it must 
be a new planet. 

It was perhaps the most startling discovery ever made in 
astronomy : Herschel named it the Georgium Sidus, in com- 
pliment to the English king, who promptly honored him 
with an appointment at court, and made him rich with a 
pension of £200. He removed to Slough, near Windsor, 
where he built "Observatory House," and made it memor- 
able as the scene of endless important astronomical dis- 
coveries. Long afterwards, Arago characterized it as u le 
lieu du monde ou il a Me fait le plus de decouvertes." 

Uranus has four satellites, two discovered by the same 
Herschel in 1787, and two by Lassell in 1851. They have 
one important peculiarity : they revolve in their orbits 
around the planet from east to west instead of west to east, 
the usual direction of orbital motion in the solar system. 
They are thus an exceptional case, and constitute in a way 
an unexplained difficulty in the Laplacian nebular hypothe- 
sis (p. 235), which would seem to require all satellites to 
revolve in the same direction. 

The outermost known planet is Neptune, remarkable 
principally on account of the interest attaching to its dis- 
covery. Shortly after Uranus had been found, astronomers 
searched their old records, and ascertained that good ob- 
servations of it existed as early as 1690. But it had always 
passed for a star, its disk not being big enough to betray its 

247 



ASTRONOMY 

planetary character on sight, even in the telescope. But no 
orbit could be found which would bring these early observa- 
tions into accord with the numerous ones which began to be 
accumulated immediately after discovery. And the planet 
soon refused to live up to its modern observations, also. 
More than one astronomer suggested that there must be an 
unknown planet exterior to Uranus., and perturbing its mo- 
tion, so as to throw it alternately in advance of its proper 
orbital position and behind it. 

In 1845, a young Englishman, Adams, who had graduated 
from Cambridge University only two years before, succeeded 
in constructing an orbit for the hypothetical exterior planet, 
basing his calculations simply on the observed discrepancies 
in the orbital motion of Uranus. He wrote to the astronomer 
royal at Greenwich, asking him to look for the new object 
with his big telescope in a certain definite position on the sky. 
We now know that this position given by Adams was correct 
within 2°, so that a little careful "sweeping" with the tele- 
scope would undoubtedly have revealed the planet. But the 
astronomer royal made an unfortunate mistake ; the story is 
that he delayed attending to Adams' letter. 

But another astronomer, Leverrier, was also working 
at the problem in France ; by August, 1846, he, too, had 
worked out the new orbit. On the 23d of September a 
letter from him arrived in Berlin and was delivered to 
Galle at the observatory there. Galle had a new and very 
complete star-chart of the proper region of the sky ; and it 
was for this reason that Leverrier had written to him, rather 
than to any other astronomer. As soon as it became dark, 
Galle went into the observatory dome, and began to compare 
his chart with the sky. He very soon found a strange body 
within less than 1° of the exact spot indicated in Leverrier's 

248 



THE PLANETS ONE BY ONE 

letter. It was an exciting moment; the new planet had 
been seen at last. 

One curious fact is that both Adams and Leverrier made 
use of Bode's law (p. 196) to obtain an approximate value 
for the supposed planet's distance from the sun. This law 
has no foundation in theory ; but it had proved to be fairly 
exact for all planets then known, including Uranus. But 
it fails for Neptune ; and accordingly both the computers were 
very largely in error as to this important element of their new 
planet's orbit. There is ground for supposing that their 
success was due in some degree to accidental favoring 
circumstances. But the result was unquestionably a great 
triumph for mathematical science and for Newton's law of 
gravitation : at this distance of time it is proper to divide 
the honor of the discovery equally between Adams and 
Leverrier. It is certainly great enough for two men, but 
in the middle of the last century, an ascerbitous controversy 
raged about the assignment of priority in this matter. 

Neptune is so far away from the earth that but few de- 
tails have been discovered concerning it. There is but one 
known satellite. And beyond Neptune, no further planets 
have been found, though the existence of such " ultra-Nep- 
tunian " bodies has often been suspected. But none has ever 
been revealed, even to the most careful photographic surveys 
so far made in the heavens. 

But there is one other material substance in the solar 
system that requires mention here. The mysterious Zodiacal 
Light is observable as a faintly luminous band traceable 
along the ecliptic circle outward from the sun for a consid- 
erable distance both east and west. There is also at times 
a faint glow called by the German name " Gegenschein " 
discernible in the part of the ecliptic opposite the sun. The 

249 



ASTRONOMY 

whole thing may perhaps be best explained as a ring of 
excessively minute planets, revolving around the sun in 
an orbit larger than that of the earth. Those near the sun 
would, of course, be the brightest ; and the Gegenschein would 
be the combined effect of an infinitude of these particles 
acting like tiny full-moons in their position of opposition 
(p. 163) to the sun. 



250 



CHAPTER XIII 

THE TIDES 

Kepler was probably the first man to notice that the 
tides of ocean are due to some form of attraction exerted 
by the moon. He looked upon the moon as a personal ally 
of the earth, and in his quaint Latin remarks that a a mutual 
affection between allied bodies tends towards their union.' ' 
It is possible that Kepler may have had some hazy idea of 
gravitation as a species of personal characteristic of celestial 
bodies. 

Let us begin by summarizing the facts easily observable 
by any one who examines the behavior of the ocean along 
its shores. In the first place, it will be found that the water 
level changes considerably. During six hours, approxi- 
mately, the waters rise ; and again, for about six hours, they 
fall. In each day there are ordinarily two high tides and two 
low tides. Furthermore, in addition to merely rising and 
falling, the water also flows along the coast, in one direc- 
tion during a period equal to the time of rising tide, more 
or less, and in the opposite direction during a period corre- 
sponding in duration to the falling tide. Thus strong tidal 
currents exist ; and navigators frequently take advantage 
of them to increase the speed of ships, especially in the case 
of sailing vessels engaged in the difficult business of beating 
(as it is called) against an adverse wind. 

We shall consider the earth for a moment as a globe 
uniformly covered with a shallow ocean. The most im- 

251 




ASTRONOMY 

portant cause of tides on such an imaginary globe would 
be the gravitational attraction of the moon. We know, 
from Newton's law, that such gravitational attraction dimin- 
ishes rapidly with an increase in the distance separating the 
attracted particle from the moon. Consequently, the moon 

attracts the water on the earth's 
surface more strongly than it 
does the more distant solid earth 
beneath it. We should there- 
fore expect the moon to heap 
up water at that point on the 
earth which is nearest to the 
moon. But there is also water on the earth on the side 
opposite the moon ; and this water is attracted less than the 
solid earth ; it is attracted least of any terrestrial material, 
because it is farthest of all from the moon. In other words 
(Fig. 64), the moon should pull the water away from the 
earth at M, tending to heap it up ; and at 0, it should pull 
the earth away from the water, again tending to heap it up. 
But the tidal forces exerted by the moon do not act in 
the above very simple way ; in fact, the actual heaping up of 
the water due to the above cause would be quite insignificant. 
A far greater effect is produced at points not directly under 
the moon. Here the tidal force is not vertical, because the 
moon is not directly overhead ; it may be regarded as divided 
between the vertical and horizontal directions. And the 
horizontal fraction is then usually the important one. It 
tends to move particles of water horizontally along the 
earth's surface, and to move them toward the place where the 
moon is overhead. But owing to the earth's axial rotation, 
the moon rises in the east and sets in the west. While it is 
east of the meridian in any given place, it is pulling the 

252 



THE TIDES 

water particles eastward with the horizontal fraction of 
the force. After crossing the meridian, it of course pulls 
them westward ; and therefore the result should be an oscilla- 
tion of the water particles backward and forward, occupy- 
ing approximately half a day to go and come. And of course 
this means half a lunar day (p. 176) ; not twelve ordinary 
solar hours. 

Since the moon, at any given moment, is east of certain 
places on the earth, and west of other places, it follows that 
different parts of the ocean will be oscillating different ways 
at the same time. This must produce a tidal wave, with 
high tide at the place where the crest of the wave is situated. 
The crest would follow the moon around, as the earth ro- 
tates, but it would not be under the moon. It can be shown 
that it would, in fact, ordinarily be 90° distant from the 
point under the moon. And there would be a second crest 
opposite the first, according to reasoning similar to that of 
Fig. 64; and therefore two daily high tides, following the 
moon around. 

Having thus outlined the explanation of the semi-diurnal 
tide, of which the period is half the lunar day, it is now 
possible to explain that the two tides each day are of un- 
equal size. In general, one rises 
higher than the other. Although /---— A\ 
we have seen that the high tidal A/^^/^V\ 
crest is not under the moon, we can I ( ^7 — -J Q j^theMo^ 
still reason as if it were, in order to \ \J^y 
explain the above inequality. As /$ " 

the earth rotates on an axis perpen- FlG - 65 - Diurnal inequality 

•.., ,, n .. of Tides. 

dicular to the plane of the terrestrial 

equator, every point on the earth's surface must rotate in a 

circle parallel to the equator. Now, in Fig. 65, supposing 

253 



ASTRONOMY 

the tidal crests to be under the moon and opposite it, and 
the earth rotating around the axis NS, a person at P will not 
have as high a high tide as he will have twelve hours later, 
when the earth's rotation has carried him around to P'. 
The difference is shown by the dotted lines at P and P', and 
is called the diurnal inequality of the tides. It is due, of 
course, to the moon's not being situated in the plane of the 
equator. If the moon were in the equator plane, the tidal 
crests would be placed symmetrically with respect to the 
equator, and the two high tides would be practically equal. 
In fact, it must happen on two days each month that the 
moon really is in the equator plane. For the moon's ap- 
parent motion on the sky, due to its orbital motion around 
the earth, appears to take place in a great circle of the celes- 
tial sphere (p. 160), which must, of course, cut the celestial 
equator at two points. And it is a fact in accord with 
actual tidal observation that the diurnal inequality disap- 
pears twice each month. Then the two tides are equal. 

To complete this part of the subject, two more details 
must be mentioned. First, we recall that the lunar orbit 
around our earth is an oval or ellipse, with the earth 
at one focus, at some distance from the center. The 
moon will therefore be especially near the earth at certain 
times. When it is at the nearest point of its orbit, the 
perigee (p. 169), its tide-raising force will be greater than at 
any other time. In fact, this force is about | greater at 
perigee than at apogee. Perigee and apogee, of course, both 
occur each month ; consequently, the high tides are by no 
means equally high during the entire month. 

The other matter requiring mention is the effect of the 
sun on the tides. It operates in a manner precisely similar 
to the moon, but its greater distance diminishes the tidal 

254 



THE TIDES 

force, so that the solar tide is only about T 5 T of the lunar tide. 
The sun is far larger than the moon, but its greater gravi- 
tational attraction due to mass or bulk is more than counter- 
balanced by its greater distance. But it is clear that when 
the sun and moon are so placed that they act together, we 
shall get especially high tides ; and when they act against 
each other, the tides will be especially feeble. 

Of course these two bodies pull together when sun, earthy 
and moon are situated more or less in a single straight line ; 
and this occurs at the dates of full and new moon (p. 163). 
We then have the great tides called Spring Tides ; and when 
the moon is in the first and third quarters we have the little 
tides called Neap Tides. Spring and neap tides have rela- 
tive heights in the ratio of 11 + 5 to 11 — 5, or 8 to 3, be- 
cause the solar pull is T \ of the lunar. 

The above brief outline of tidal theory is greatly modi- 
fied when applied to the actual earth, upon which the 
oceans are deep bowls, large, but still limited in size; 
and the gulfs, sounds, etc., small, shallow limited cups. The 
laws of wave motion in areas limited as to size and depth 
come into play : according to these laws, the rate of progress 
of a wave, or its time of oscillation, depends on the depth 
of water. For instance, in a basin like the north Atlantic 
Ocean, a wave would move 500 miles per hour if it were 
set in motion, and then left to itself. It would pass from 
Europe to America in about six hours. Thus its period for 
going and returning would be nearly equal to the tidal 
period of half a lunar day. It can be demonstrated that 
when these two periods are thus about equal, the tides will 
be large. The water will practically oscillate about a 
neutral line in the middle of the ocean, giving high tide at 
the European coast when it is low tide at the American. 

255 



ASTRONOMY 

But this explanation is complicated still further by the 
configuration of the coasts, whereby the Atlantic does not 
act as a single basin with a single neutral line, but as several 
basins overlapping, more or less. But the general result is 
nearly as stated. 

When we come to the peculiar tides belonging to limited 
areas of the coast, — such a basin as Long Island Sound, for 
instance, — still a different explanation is required. Here the 
tidal wave is not really due directly to the moon ; it is a special 
local oscillation, set up by contact or impact from the 
lunar tide in the ocean outside the sound. In such cases, 
conditions become quite complicated, and often lead to 
tides much higher than the ocean tides. For instance, in Long 
Island Sound the rise and fall is about seven feet ; high tide 
occurs at nearly the same time throughout the sound ; and 
the wave motion produces a rapid current, or motion of the 
water particles along the sound. Another interesting tidal 
modification is found in the funnel-shaped Bay of Fundy, 
where the tides rise and fall as much as forty feet, or even 
more. 

Tidal phenomena produce results of importance other than 
recurrent changes in the oceans. Tidal evolution is a term 
used to describe effects produced on the earth as a whole by 
tidal action continued throughout vast ages of time. It 
is clear that the tidal motions of great masses of water must 
consume a vast quantity of mechanical energy, especially 
where the great tidal currents occur along the ocean coasts. 
In such cases there must also be much friction between the 
land and water. Friction will generate heat, and consume 
more energy. 

Now all this energy must be derived from some source : 
the law of the conservation of energy (p. 2) tells us that there 

256 



THE TIDES 

can be no manifestation of new energy, such as we have 
just mentioned, without an equal and corresponding dim- 
inution of the manifestation of energy somewhere else. 
The place where we should expect to find this diminution 
is in the earth's rotation. In other words, we should expect 
tidal friction, etc., to act as a sort of brake on the earth's 
axial rotation, and to bring about a consequent minute 
lengthening of the terrestrial day, after the lapse of sufficient 
ages of time. But the most delicate astronomic observations 
have failed to detect any such lengthening of the day. It 
must therefore be extremely small, certainly not more than 
to oo~o~ °f a second in a century. 

But there is another interesting consequence of these con- 
siderations : how does terrestrial tidal friction affect the 
moon's motions? In Fig. 66, the moon is shown in the 
celestial equator, and the 
two great tidal protuber- 
ances at H and H f . Ac- 
cording to theory, as we 
have seen (p. 253), these 
protuberances or tidal 
crests should be at A and 
B, 90° from the point 

, _ Fig. 66. Tidal Effect on the Moon. 

under the moon. But 

so much of the tidal effect as acts like friction, to retard 
the terrestrial rotation, must also make the two protuber- 
ances lag behind their proper positions at A and B. This 
brings H nearer the moon than W ; increases the lunar 
attraction at H, as compared with H' ; and therefore ac- 
celerates the moon's motion in its orbit, as shown by the 
arrow. 

Now it can be demonstrated from the principles of 

s 257 




, ASTRONOMY 

mechanics, that increasing the velocity of a body moving in 
an orbit will increase also the size of the orbit and the period 
of revolution of the body in the orbit. Therefore, tidal 
friction must make the moon recede from the earth, and 
must also lengthen the lunar sidereal period. In the hands 
of G. H. Darwin, these simple principles have led to an 
extremely plausible theory as to the formation of our moon. 
According to Darwin, the moon once formed part of the 
earth ; the entire mass was in a semi-liquid or plastic con- 
dition ; and was in quite rapid rotation about an axis. There 
was a tremendous flattening of the earth at the poles, due 
to plasticity. It can be shown mathematically that such a 
rotatory flattened plastic body may assume any one of 
several shapes. One of these possible figures is pear-shaped. 
The fact that we have a moon is thought by Darwin to 
prove that the pear-shaped figure actually was the one that 
happened to prevail. The rotating pear-shaped figure should 
then pass over into an hour-glass ; from that to a dumb-bell, 
with unequal weights at the ends. Finally comes a separa- 
tion; a true planet with a moon, both revolving rapidly 
about their common center of gravity, and very near each 
other. 

Now come gigantic tides ; tides compared with which our 
present ocean tides are absolutely insignificant. For the 
plastic earth was subject to great bodily tides, not merely 
little oscillations of a thin shell of ocean. Frictional forces 
then produced no mere slight perturbative action ; they were 
dominating forces. The moon was driven farther and 
farther from the earth; and the lunar sidereal period was 
lengthened, until both bodies reached the condition now 
existing. 

If this theory is correct, it enables us to predict for future 

258 



THE TIDES 

ages the final condition of our moon, when the last stage of 
equilibrium shall finally prevail. When that occurs, the 
lunar sidereal period and the terrestrial day will be equal, 
the earth rotating on its axis in 55 of our present days, and 
the moon making an orbital revolution around the earth 
in precisely the same period. The moon should then be 
always opposite the same point of the earth's surface ; and 
both bodies should revolve as though both were attached 
rigidly to the ends of an unbending bar. 



259 



CHAPTER XIV 



THE SOLAR PARALLAX 



We have had occasion to mention several times the im- 
portance of a correct knowledge of the distance separating 
the earth from the sun. In our discussions of planetary 
motions we have considered this distance to be known; 
in fact, we have assumed all the elements (p. 200) of the 
earth's orbit to be within our knowledge. 

Until the latter part of the eighteenth century, astronomers 
had only a very rough knowledge of the sun's distance, or 
of its Parallax. This last term may be defined easily ; it is 
exactly analogous to the corresponding term already defined 
(p. 169) in the case of the moon. By the solar parallax we 

simply mean half 
the angular diame- 
ter (p. 118) of the 
earth, supposed to 
be seen from the 
sun. Thus, in Fig. 67, imagine an observer situated on the 
sun at S. Draw a straight line from S to the center of the 
earth at C and another to the surface of the earth at 0. 
Then the angle at S between these two lines is half the angu- 
lar diameter of the earth as seen from the sun, and is there- 
fore the solar parallax. A simple equation exists, 1 by means 

1 If, in Fig. 67, we let D represent the sun's distance ; tt, the parallax 
angle ; and r the earth's radius ; we have at once, from the triangle SCO : 

tan ir = i or D = r^—' (Of. Note 20, Appendix.) 
D tan 7r 

260 



-JT 




Fig. 67. Solar Parallax. 



THE SOLAR PARALLAX 

of which we can calculate the solar parallax from the solar 
distance, or the distance from the parallax. If either be 
known, we can at once find the other. So the term " solar 
parallax" is really a substitute for " solar distance." The 
two terms are interchangeable, in a way; but they are 
not synonymous. One is an angle, the other a linear dis- 
tance. The present accepted value of the solar parallax is 
8."80. 

We shall now consider various ways of measuring it. 
The reader will remember the method already mentioned 
(p. 168) for ascertaining the moon's distance by simultaneous 
observations of that body from two observatories, one in a 
high northern latitude on the earth, and the other in a high 
southern latitude. Of course this same method might be 
applied to the sun, but there is an objection that renders it 
almost useless. This objection arises from the small size 
of the parallax angle, which is really the quantity to be 
measured. The lunar parallax is about 1°, the solar only 
8. "8; consequently, any small error of observation, such as 
one-tenth of a second of arc, will have a considerable effect 
in the case of the solar parallax, while it would be quite in- 
appreciable in the case of the moon. 

This difficulty can be obviated in some degree by measur- 
ing the solar distance in an indirect manner. The distance 
is really only used to ascertain the scale, or size, of the 
planetary orbits. For with the aid of Kepler's harmonic 
law (p. 188), we can find the relative distances of the various 
planets from the sun, after we have observed their periods. 
Then, knowing the distances, we can make a map of all the 
orbits, here once more supposed to be concentric circles. 
And, again with our knowledge of the periods, we can locate 
the planets themselves in their orbits on the map, for any 

261 



ASTRONOMY 

date, if we have also observed the dates of conjunctions, 
etc. (p. 209). Such a map will be correct in every respect, 
except that the scale remains unknown. The heavenly 
bodies will be shown in their proper relative places on the 
date in question, but we do not know the number of miles 
corresponding to an inch on the map ; in other words, the 
scale of the map. To ascertain this, it will be sufficient to 
measure observationally the distance from the earth to any 
other planet on the date for which the map was drawn. This 
distance once known from the observation in miles, every 
other distance on our map of the solar system also becomes 
known in miles. 

This work will be most accurate, if we select for measure- 
ment a planet which comes comparatively near the earth, 

and make our observations 
and our map at the time of 
closest approach. For, 
after all, distance from the 
earth can be measured only 
by using the earth's own 
diameter in the way sur- 
BaSe ' li D\ f t arSe 0mpUtillg Planet ' S veyors use what they call a 

" base-line." Thus, in Fig. 
68, for a planet at P, we can at best only measure the angles 
POM and PMO, so as to ascertain the planet's distance by 
constructing the triangle POM from the known base OM (the 
earth's diameter) and the two measured angles. 1 But the 
base OM is always necessarily wofully short, compared with 
the planet's distance; the slightest errors in the measured 
angles produce very large errors in the distance. Therefore 

1 We can, of course, substitute a solution of the triangle by trigonometry 
for the geometrical construction. 

262 





THE SOLAR PARALLAX 

we must do this work when we can observe a planet that 
is as near to us as planets are ever found. 

The planet Mars has been used with advantage. A 
time is selected when Mars is in opposition (p. 212) so that it 
comes to the meridian at midnight, and can therefore be 
observed almost all night. And an opposition 
is chosen, too, when the earth has one of its 
closest approaches to Mars. This combina- 
tion of conditions gives the most favorable 
state of affairs for the desired measurements. , 

Fig. 69. iavor- 

Figure 69 shows the positions of the sun, earth, able Opposition 

OT IVtflT'S 

and Mars at the time of opposition. The 
orbits are not circles, and therefore the distances of the two 
planets from the sun are variable. If the opposition is one 
at which the earth happens to be at its greatest distance 
from the sun (aphelion), and Mars at its closest possible 
approach to the sun (perihelion, p. 120), the distance between 
the earth and Mars is as small as it can ever be ; and the con- 
ditions are especially favorable for its precise measurement. 
Having thus secured a favorable opposition, there are two 
different ways in which the Martian distance can be observed. 
We may employ a modification of the method already 
described for the moon (p. 168), and observe Mars from 
two terrestrial observatories situated as far apart as possible. 
In that case our base-line is the line joining the two ob- 
servatories. Or we can use the "diurnal" method. In this 
method, the planet is observed from the same place on the 
earth at two different times on the same night : first, shortly 
after sunset ; and second, shortly before sunrise. In the 
interval, the rotation of the earth on its axis will carry the 
observer to a different position in space ; and the line joining 
his two positions becomes the base-line. 

263 



ASTRONOMY 



Thus, in Fig. 70, let us disregard the slow orbital motions 
of Mars and the earth, since these will amount to but little 
in the few hours elapsing between the two observations ; 
and consider the earth's diurnal rotation only. Let the 
earth's center be at C, with the rotation axis passing through 




Fig. 70. Diurnal Method. 

C perpendicular to the printed page; and the observer at 
O w . Then the observer will be carried in about ten hours 
from O w to O e by the diurnal rotation, and the length of the 
line O w O e can be calculated easily from the known dimen- 
sions of the earth, and the time elapsing between the two 
observations. This line O w O e becomes the base-line. From 
the point O w we see the planet projected on the celestial 
sphere in the direction MW ; from O e we see it in the direc- 
tion ME. The difference of the two measured directions is 
the angle O w MO e ; from this, together with the known base- 
line O w O e , we can calculate the distance MC from Mars to 
the earth. 

In making the observations, both for the diurnal method 
and for the method with two observatories, the most accurate 
way to observe is to use as an auxiliary some 
small star appearing near Mars on the sky. 
In Fig. 71, which represents a part of 
the sky, such a star is shown. In either 
method of observation, owing to the ob- 
server's change of position from one end of 
the base-line to the other, the observations show the planet 

264 




5-tar 

Fig. 71. Observa- 
tion of Mars. 



THE SOLAR PARALLAX 

projected at two slightly different positions on the sky, as 
Mi and M 2 . The star itself is always seen in the same posi- 
tion, because the stars are all practically infinitely distant 
in comparison with any base-line available on the earth. 

The angular distance on the sky between the star and Mars 
(or its equivalent, the difference in direction of the star and 
Mars as seen from the earth) can be measured in the same 
way that the angular diameter of a planet is measured 
(p. 203), with an instrument called a Micrometer, to be 
described later. 

Thus we obtain from the two observations the angular 
distances of M x and M 2 from the star. Their difference is 
the arc MiM 2 on the sky ; and this is the angle O e MO w of 
Fig. 70, or the angle subtended by the base-line O e O w at 
the distance of Mars from the earth. Comparatively simple 
calculations will then transform this angle into a knowledge 
of the Martian distance, the Martian parallax, and thence 
the solar parallax. 

The diurnal and the two-observatory methods each have 
advantages and disadvantages. In the diurnal method, the 
observations can all be made by one man in one place with one 
instrument. This eliminates those errors that arise from per- 
sonality of the observer, and differences between different in- 
struments, etc. Gn the other hand, the two observations are 
necessarily separated by several hours in time, while they can 
be made quite simultaneous in the two-observatory method. 
In our present discussion, we have neglected totally the slight 
orbital motions of Mars and the earth. This is without 
effect if the two observations are simultaneous ; but they 
never are so in the diurnal method. The two planetary 
motions must be taken into account by calculation ; and 
thus any slight existing errors in our supposed knowledge of 

265 



ASTRONOMY 

the two planetary orbits produce slight indirect inaccuracies 
in the resulting parallax determination. 

But in the two-observatory method it is by no means 
easy to secure perfectly simultaneous observations, either. 
The two stations are very far apart on the earth. The 
weather is quite likely to be cloudy at one station when it 
is clear at the other, thus preventing simultaneous results. 
Indeed, vagaries of weather sometimes seem especially 
designed to hinder astronomers in their work, particularly 
when simultaneous observations are required. But in the 
diurnal method, the astronomer carries his weather with him, 
as it were, while he is rotated by the earth from one end 
of his base-line to the other. If he begins with a good clear 
night, he is quite likely to secure the necessary corresponding 
second observation. 

The best measurements of Mars by the diurnal method 
were made by Gill in 1877. He organized a special astro- 
nomical expedition to the island of Ascension in the south 
Atlantic ; and this was an especially favorable spot for his 
purpose. It was desirable to be near the equator, so that 
the diurnal base-line might be a long one. For at the pole, 
of course, the diurnal circle shrinks into a mere point. 

Gill obtained the value 8. "78 for the sun's parallax from 
the Ascension expedition, but it appeared that various 
causes interfered to render the result less exact than was 
desired. Chief among these causes was the difficulty of 
measuring the angular distance between the planet and a 
neighboring star in the manner we have described. This 
difficulty arises from the fact that Mars appears in the 
telescope as a disk, while the stars, of course, show only 
tiny points of light : and there seems to be some kind of 
personal error introduced by the effort to measure from a 

266 



THE SOLAR PARALLAX 

disk to a point. For this reason, Gill decided to repeat 
the work, using certain of the planetoids (p. 231) whose 
orbits are located between Mars and Jupiter. These little 
planetoids, of course, appear in the telescope like star 
points, and the above cause of personal error does not 
arise. 

Having been appointed director of the great observatory 
maintained by the British admiralty at the Cape of Good 
Hope, Gill attacked the problem on a large scale, using three 
different planetoids, Iris, Sappho, and Victoria, all of which 
have orbits suitably situated for the purpose. He caused 
to be constructed a special instrument for measuring the 
angular distances between the planetoids and neighboring 
small stars. This instrument is called a Heliometer. Four 
were made, and mounted respectively at the Cape of Good 
Hope, New Haven, Leipzig, and Gottingen. With all 
these special instruments simultaneous observations were 
made in such a way that both the diurnal and the two-ob- 
servatory methods could be used in the subsequent calcula- 
tions. The final result of the whole campaign was to fix the 
solar parallax at 8. "80 : this value is now regarded as the 
best, and has been adopted by all authorities to determine 
the scale of the solar system, and to perform calculations of 
every kind relating to planetary motions. 

Since this work of Gill's was completed, a certain newly 
discovered planetoid, Eros (p. 236), was found to have 
an orbit so placed that it can at times approach the earth 
nearer than any other object in the heavens except our own 
moon. Consequently, its distance from the earth must 
admit of very accurate determination. One of the close 
approaches of Eros, or favorable oppositions (cf. p. 263), 
occurred in 1900 ; and extensive observations were then made, 

267 




ASTRONOMY 

this time by the newly perfected method of photographic 
observation. Results have been published only very re- 
cently, and they confirm Gill's value, 8. "80. 

This method of minor planet observation is so superior 
that all other methods are of historic interest only; but, 
historically, the famous determinations from transit of 
Venus observations (p. 221) are well worth a careful study. 
If we consider the motion of an inferior planet like Venus, and 
assume the orbits of both Venus and the earth to lie in a 
single plane, then, at each inferior conjunction, Venus will 
pass between the earth and the sun. This 
is evident from Fig. 72, which once more 
shows Venus at inferior conjunction. In 
point of fact, the orbits do not lie exactly in 
a plane, but, nevertheless, a passage between 
us and the sun does sometimes occur. It 

Fig. 72. Venus at 

inferior Conjunc- will happen whenever the inferior conjunc- 
tion takes place at about the time when 
Venus is at one of the nodes (p. 200) ; or, in other words, 
when the conjunction happens while Venus is on the line of 
intersection of the two orbit planes of Venus and the earth. 
When on this line of intersection, Venus is for the moment 
in both planes ; and if there is also an inferior conjunction 
at the same time, there must be a transit. 

Venus, during transit, is seen as a small round black dot 
projected on the bright disk of the sun. This dot appears 
to enter the solar disk on the western edge, transits the 
sun in a line approximately straight, and finally passes away 
from the sun again at the eastern edge. It then disappears ; 
for, of course, we cannot see Venus against the sky back- 
ground, when near the sun, since the illuminated side of the 
planet is then turned toward the sun. We see the dark side ; 

268 



THE SOLAR PARALLAX 

it is "new Venus," if we may borrow a term from the analogy 
of the moon. 

Halley, a famous astronomer royal of England, showed 
how to determine the solar parallax from transit of Venus 
observations. 1 His method is shown in Fig. 73. 




Fig. 73. Transit of Venus, Halley 's Method. 
(After Herschel's Outlines of Astronomy, London, 1851, p. 289.) 

Two observers on the earth, located at the points A and B, 
widely separated in latitude, are provided with good clocks, 
and observe the exact quantity of time required by Venus 
to complete a transit across the sun's disk. But these two 
observers will see Venus crossing the sun along two different 
lines or " chords" SP and sp. The lengths of these two 
chords can be calculated in seconds of arc ; and thence the 
solar parallax can be determined. 2 

Halley thought it would be possible to observe the dura- 
tions belonging to the two chords within two seconds of 
time. In actual observable transits of Venus this would give 
the solar parallax correct within one-hundredth of a second 
of arc. But no such precision of observation has ever been 
possible, principally because Venus has an atmosphere 
(p. 220) which introduces errors in the observed durations 
of the chords. Even in the modern transits of 1874 and 
1882. these errors were as great as 10 seconds of time. 

1 See Phil. Trans. Roy. Soc. Lond., Vol. XXIX, p. 1716, or Hutton's 
Abridgment, Vol. VI, p. 243. 

2 Note 33, Appendix. 

269 



ASTRONOMY 

There is an interesting astronomical story connected with 
the first observed transit of Venus. It seems that in the year 
1639 there was a young curate in England ; a man living in 
miserable circumstances, but nevertheless inspired with an 
extraordinary zeal in the study of astronomy. This man, 
Horrocks by name, and at the time but 22 years of age, 
united in his own person two of the most poorly paid pro- 
fessions, that of an unbeneficent clergyman of the estab- 
lished church, and an astronomer. A diligent student 
of Kepler's writings, he had been able to correct an error of 
the latter, and to predict that a transit of Venus would 
occur on Nov. 4, 1639. He was unable to fix the exact 
hour, but he had a little telescope, and prepared him- 
self to watch the sun through the entire day. 

Now comes the peculiarly human part of the story. He 
found that the eventful date would occur on a Sunday. It 
seemed of the last importance to secure the observation, 
which was at that time an unprecedented one ; the circum- 
stances therefore found him undecided between his duty at 
church and his keen desire to secure fame as an astronomer. 
His sense of duty prevailed : he decided to give to the tele- 
scope only the intervals between services. And he had his 
reward, after afternoon service. Hurrying to his poor home, 
he was in time to see the black round planetary dot on the 
sun just before sunset, which happens at a very early hour 
in the northern latitude of England, and in November. 
To-day, a tablet may be seen in Westminster Abbey, 
bearing a Latin inscription commemorating this famous 
observation. 1 

1 A very good account of it is to be found in Cassini's Elements 
d'Astronomie, published in 1740. At that time the Horrocks observa- 
tion was still unique, and Cassini founds many calculations upon it. The 
tablet in Westminster bears a quotation from Horrocks' own work, Venus 

270 



THE SOLAR PARALLAX 

Having thus outlined several methods of determining 
the sun's parallax by observations of planets, it remains 
to mention certain indirect methods of arriving at its value. 
For instance, the sun's distance can be computed from 
the theory of the aberration of light (p. 136). * Another 
quite independent way is called the perturbation method. 
Briefly stated, it consists in measuring the slight perturba- 
tions (p. 403) produced in the regular elliptic motions of 
the planets. These perturbations are caused by gravita- 
tional attractions between the bodies concerned, and the 
mathematical equations expressing them involve the distance 
from earth to sun as a factor. If this distance is known, it 
is possible to compute the perturbations ; or, the perturba- 
tions being measured observationally, the solar distance may 
be computed. 

in Sole Visa, published by Hevelius in 1662 at Dantzig. The quotation 
reads, "Ad majora avocatus quae ob haee parerga negligi non deeuit." 
1 Note 34, Appendix. 



271 



CHAPTER XV 

ASTRONOMIC INSTRUMENTS 

It is not easy to understand the details of instruments 
from printed descriptions and illustrations. A short verbal 
explanation, by an astronomer in an observatory, with the 
instrument under discussion before him, is the very best way 
to gain an insight into the methods and machinery of obser- 
vation. For those who have no opportunity to visit an 
observatory, we give here a brief account of the most impor- 
tant kinds of astronomic apparatus, prefacing it with Plate 12, 
a photograph of the famous Lick observatory buildings on 
the summit of Mt. Hamilton, in California. 

To begin with the telescope itself. In the popular imag- 
ination, it is a big tube more or less filled with lenses from 
end to end. But this notion is quite wrong. Theoretically, 
the telescope has two lenses only, one at each end of the tube. 
The large lens, which is turned toward the sky, is called 
the Object Glass : upon it falls the light coming from the 
celestial object under observation. This light is concen- 
trated or "focuse'd" by the object glass, and forms an 
image of the celestial body near the small end of the tube 
where the observer places his eye. Between this focal 
image and the object glass, the tube is empty. 

The other telescope lens is placed at the small end of the 
tube, between the observer's eye and the focal image, but 
very near the latter. This lens is simply a magnifying glass, 
or microscope, and is intended to enlarge the focal image, so 

272 



ASTRONOMIC INSTRUMENTS 

that the observer will see more detail than would be possible 
with the eye alone. This eye-end lens of the telescope is 
called the Eye-piece. In modern instruments, both telescope 
lenses are of the kind called " Compound" lenses. Each 
is made up of two or three separate lenses, placed close 
together, or even in actual contact. By this compounding 
of the lenses it has been found possible to eliminate partially 
certain optical imperfections from which all lenses suffer. 
But each compound lens really acts like a simple lens, except 
that it does its work better. Galileo's telescope of 1610, 
which found the moons of Jupiter and the spots on the sun, 
had single lenses only. 

Telescopes intended for terrestrial use have an extra lens in 
the eye-piece, called an " erecting" lens. For the simple 
astronomic telescope reverses the image of the object we look 
at ; we see it with the top and bottom interchanged, and the 
right and left sides likewise inverted. This is of no conse- 
quence in astronomy, since there is no up or down in space, 
and a round planet may be observed just as well one way as 
the other. But for terrestrial purposes, we must have ob- 
jects represented to the eye in their true positions. This 
extra erecting lens diminishes slightly the efficiency of the 
telescope, because it introduces two additional glass surfaces, 
the two sides of the erecting lens itself. And as human 
hands cannot grind lenses with absolute accuracy to their 
correct theoretic shape, it follows that the erecting lens 
causes slight errors that do not exist in the astronomical 
eye-piece. Plate 3, p. 17, shows the moon as seen in an 
astronomical or inverting telescope (cf. p. 166). 

The question is often asked: "What is the magnifying 
power of a given telescope?" Or the same question occurs 
in another form : * ' How near does this telescope bring the 
t 273 



ASTRONOMY 

moon?" These two questions are really one. The moon's 
distance is 240,000 miles (p. 169) ; a telescope magnifying 
1000 times would therefore bring it within a distance of 240 
miles ; or give us as good a view, approximately, as we would 
get with the unaided eye if the moon were only 240 miles 
away. 

From what has been said before, it is perfectly clear that, 
within certain limits, the magnifying power of a telescope 
is just as great as we care to make it. The magnifying power 
comes from the power of the eye-piece regarded as a mi- 
croscope : it is evident that we can use a microscope of high 
power, or one of low power, on the same telescope, at dif- 
ferent times; and thus we can vary the magnification af- 
forded by the instrument as a whole. But there is a definite 
practical difficulty that limits the available power of the 
eye-piece. 

Suppose we are observing a planet, such as Mars, for in- 
stance. The quantity of light received from Mars may be 
regarded as constant, and therefore a constant quantity of 
light from Mars reaches the focal image. This light is there 
spread uniformly over the surface of that image. Now if 
we double the magnifying power of the eye-piece, we shall 
see twice as large an image. The same quantity of fight 
from Mars is therefore spread over a larger surface, and so 
the image is dimmer than before. Increase of magnifying 
power in the eye-piece enlarges the image and brings out 
more detail; but it makes that detail fainter (cf. p. 230). 

If we continue to increase the magnification, there must 
come a time when we shall increase the detail, but will be 
unable to see it on account of faintness. For these reasons, 
astronomic telescopes are provided with a " battery" of 
eye-pieces of different powers. It is customary for the 

274 



ASTRONOMIC INSTRUMENTS 

astronomer to try gradually increasing powers until he finds, 
by experiment, the one that gives the best result. It will not 
be the same one each night, because a little higher power 
than usual may be employed when the terrestrial atmosphere 
is especially clear. 

As soon as the above limit of power is reached, no further 
increase is possible, unless we enlarge the object glass ; or, 
in other words, make the whole telescope bigger. With a 
larger object glass we can gather more light from Mars, 
because the '4ight-gathering" power of an object glass 
^rnust increase with an increased area of the glass itself. 
And if we gather more light, we can have a larger focal 
image without making it too dim for practical use. Ex- 
perience has shown that under the most extremely favorable 
terrestrial atmospheric conditions, it is possible to use a 
magnification of about 100 for each inch in the diameter of 
the object glass. In the case of the great Yerkes telescope, 
the diameter of that glass is 40 inches; a power of 4000 
should therefore be conceivably possible; and the moon 
should be brought within the equivalent of ^-f^HS"^, or 60 
miles. But this theoretic result is never quite attained in 
practice, because all imperfections of the atmosphere are 
magnified by increased optical power. We see the moon as 
we would see it with the unaided eye at a distance of 60 miles ; 
but through an atmosphere more like water than air. 

Among the most important accessories of a telescope, when 
it is to be used for accurate measurement, is a pair of " cross- 
threads" at the focus. These threads are usually made of 
spider web ; for they must be extremely delicate, so that 
magnification by the eye-piece will not prevent accuracy of 
observation. The field of view of a telescope provided with 
cross-threads would look like Fig. 74. When the focal image 

275 




ASTRONOMY 

of a star is brought to the exact intersection of the cross- 
threads, by moving the telescope, the instrument is aimed or 
" pointed" accurately at the star; and if the telescope is 
attached to brass circles divided into degrees 
and minutes, it is possible to measure the exact 
direction in which we see the star projected on 

Fig. 74. Cross- # . 

threads in a the celestial sphere. This kind of measure- 
ment is fundamental in the astronomy of 
precision (cf. p. 197). 

Sometimes the single pair of fixed cross-threads is replaced 
by a pair of parallel threads aa' and W shown in Fig. 75, 
together with a cross- thread cc' . In such an arrangement 
the two parallel threads aa' and W are made movable, while 
cc' is fixed. The two parallel threads can be moved nearer 
together or farther apart by means of suitable screws out- 
side the tube of the telescope ; and a method is also provided 
for measuring accurately their distance asunder. This ar- 
rangement is called a micrometer (p. 265), and with it short 
angular distances on the sky can be measured with very high 
precision. Examples of such measurements are the obser- 
vation of Mars to ascertain the solar parallax 
(p. 265), the angular diameter of the sun (p. 118) 
or of the planets (p. 203), etc. 

The methods of mounting telescopes for as- <r-^b 
tronomical purposes are most interesting. It is Micrometer. 6 
of course essential that the tube be movable : it 
must be possible to turn it about pivots or "axes," in order 
that it may be pointed toward different parts of the sky. 
The most simple form of mounting is indicated in Fig. 76, 
which shows an instrument called a Meridian Circle. OE 
is the telescope, being the object-glass and E the eye-piece. 
At / is the focus, containing the cross-threads. AX is a 

276 




ASTRONOMIC INSTRUMENTS 



rotation axis, firmly attached to the telescope. There is no 
motion of the instrument, except rotation around this one 
axis; but a complete rotation about that axis is possible. 
The line AX is made to point due east and west when in 
proper adjustment; and 
it is made perfectly 
level. So the telescope 
must point north or 
south accurately, since 
it is placed at right 
angles to the axis AX. 
It follows that if the 
" sight-line " EfO be con- 
tinued outward indefi- 
nitely beyond 0, until 
it reaches the celestial 
sphere, it will meet that 
sphere at some point of 
the celestial meridian 
(p. 36). And if the 
telescope is turned 
through a complete 
rotation around the 

axis AX, the sight-line EfO may be imagined to trace put 
the celestial meridian on the sky. 

It results from these considerations that the meridian 
circle can observe stars on the celestial meridian only; 
and, conversely, if a star be observed on the cross-threads of 
the telescope, the observer knows that it is at that moment 
projected on the celestial meridian. If the exact time of the 
observation be noted, too, it is possible to calculate the right- 
ascension (p. 34) of the star ; and thus are the right-ascen- 

277 




Fig. 76. Meridian Circle. 



ASTRONOMY 

sions of stars and planets determined observationally by 
astronomers. 

This meridian instrument is provided also with two brass 
circles c and c', divided into degrees and minutes of arc. 
By the aid of these circles it is possible to measure the altitude 
(p. 36) of the star, or its angular elevation above the horizon 
at the moment when it is observed to be on the meridian. 
From this measurement of altitude it is possible to calculate 
the star's declination (p. 34) . Thus the meridian circle makes 
known both the right-ascension and declination of the star 
or planet, and these give us its exact location on the sky at 
the moment of observation. 

It will be noticed that a precise record of the time of such 
observations is most essential. For this purpose astronomers 
employ " standard" pendulum clocks of the most extreme 
accuracy, usually kept in vaults and air-tight cases where 
the temperature and barometric pressure are not allowed 
to vary, so as to produce inaccuracy in the running of the 
clocks. For the actual record of the time of observation, 
the clock is connected with an electric " chronograph." This 
instrument maintains an automatic record of the running 
of the clock upon a sheet of paper attached to a revolving 
brass drum; and upon this same sheet the observer can 
record electrically the instant of time when he makes his 
observation ; and he can make this record without his error 
ever exceeding one-fifth of a second of time. By taking 
the mean of several observations, the average error can even 
be reduced below this small amount. 

The above process, as outlined, indicates the method of 
observing stars of unknown location on the sky in order to 
make known their right-ascensions and declinations. But 
the same meridian telescope, clock, and chronograph can 

278 




PLATE 13. The Lick Telescope. 



ASTRONOMIC INSTRUMENTS 

be used for the observation of known stars ; and will then 
furnish a check upon the time indicated by the standard clock. 
It is by this latter process that the astronomic standard 
clocks are kept correct in order that the observatories may be 
able to distribute correct time telegraphically for the control 
of the " regulator" clocks that are to be found in most 
jewelers' shops, where people "set" their watches (cf. p. 18). 

Having thus described briefly the astronomer's instrument 
of precision, the meridian circle, we must next consider the 
" equatorial" mounting, the arrangement with which almost 
all ordinary telescopic observations are made. This is 
the form of mounting usually fitted with a micrometer for 
the measurement of small angular diameters, distances, etc. 
A photograph of a large telescope, mounted equatorially, is 
reproduced in Plate 13. This instrument is set up at the 
Lick observatory; the diameter of the object-glass is 36 
inches ; and the whole observatory floor is built like an ele- 
vator, so that it can be moved up and down, to accommodate 
the observer when the tube is directed to various parts of the 
sky. The supporting pillar of the telescope mounting 
passes down through a hole in the floor, so that the in- 
strument itself is not disturbed, when the floor is raised or 
lowered. James Lick, donor of the. telescope, is buried 
under it. 

The first essential of such a mounting is some form of 
" universal joint," so that the tube may be aimed at any part 
of the sky. A single axis, such as that of the meridian circle, 
is not sufficient. Accordingly, in the equatorial, shown 
again in Fig. 77, we find two axes, A and A' y perpendicular 
to each other. The telescope can be rotated around the axis 
A' ; and this axis itself, with the telescope attached, can be 
rotated around the axis A. A combination of the two rota- 

279 



ASTRONOMY 



tions furnishes a universal motion, giving access to any 
part of the sky. By means of these rotations, the tube can be 
moved from the position of Fig. 77 to that of Plate 13. 

The axis A is called the polar axis ; and the instrument is 
so constructed and adjusted that this axis points directly 

toward the celestial pole 
(p. 32). Since the stars 
perform their apparent 
diurnal rotations (p. 33) 
around that pole, it fol- 
lows that they will seem 
to perform them around 
the polar axis of the 
equatorial. This sim- 
plifies the use of the in- 
strument; for a star 
once brought into the 
field of view of the tele- 
scope, we can keep it there 
by moving the instru- 
ment around the polar 
axis only. For the tele- 
scope must be kept mov- 
ing to prevent the diur- 
nal rotation of the stars 
from carrying the object under observation out of the field of 
view. The necessary rotation around a single axis can be 
accomplished automatically by means of clock-work, shown 
inside the vertical supporting pillar in Plate 13. The as- 
tronomer is thus left free to pursue his observations without 
any further attention to the telescope. If there were no 
inclined polar axis, but, in its place, a pair of axes, one verti- 

280 




Fig. 77. The Equatorial. 




PLATE 14. The Crossley Reflector. 



ASTRONOMIC INSTRUMENTS 

cal and the other horizontal, this simple clock-work plan 
would be impossible. 

The equatorial carries two circles, c and c', divided into 
degrees, and attached to the axes A and A'. With the circle 
c' it is possible to measure the declination (p. 34) of an object 
in the field of view ; and with the circle c its hour-angle (cf . 
p. 68) can be measured. And if we wish to find a known 
object which is invisible to the unaided eye, we have merely 
to turn the telescope around the two axes, until the two 
circles indicate the object's known declination and hour- 
angle, when it will at once appear in the field of view. 

The foregoing description of the telescope applies to the 
" refractor" with its object-glass and eye-piece. But there 
is another form of instrument, the " reflector," in which the 
object-glass lens is replaced by a curved mirror. This 
forms a focal image similar to that given by a lens ; and the 
image is again examined with an eye-piece lens as before. 
Plate 14 shows the equatorially mounted Crossley reflector 
of the Lick observatory. The polar axis is shaped like a 
double cone ; and the reflector is at the bottom of the big 
tube. The focal image is formed near the top of the tube, 
and is there examined with an eye-piece shown in the plate. 
It was with this instrument that Keeler made his famous 
photographs of spiral nebulae, one of which is shown in Plate 
2 (p. 4) . Others will be described in a later chapter. 

Telescopes intended for astronomic photography are 
always mounted equatorially, and their clock-work mech- 
anism is made especially precise. For when it is desired 
to make long photographic exposures, the telescope, which 
takes the place of a camera, must of course be kept in motion, 
so as to neutralize the diurnal rotation of the stars. If this 
is not done quite exactly, we obtain only a worthless " moved 

281 



ASTRONOMY 

negative." To be certain of this essential element in astro- 
photography, such telescopes are usually made with two 
tubes, like a pair of opera-glasses. The one tube is a photo- 
graphic telescope, the other a visual one, provided with 
cross- threads. With this arrangement, the astronomer can 
watch an object with the visual telescope, while it is being 
photographed in the other. 

If the clock does not move quite perfectly, the error will 
at once show itself ; for the object will move away from the 
cross-threads in the visual telescope. The slightest tendency 
to such motion must be prevented ; and for this purpose the 
clocks of such instruments are provided with certain adjust- 
ing screws, or other devices, with which the astronomer can 
correct any possible errors of the clock while it is actually 
running, and while the photograph is being exposed. 

The last important astronomic instrument to be described 

here is called the Spectro- 
scope, of which Fig. 78 
shows the essential parts. 
The reader will recall 
that if we look at a 
source of light through 
a glass prism, it will be 
spread out into a band 

Fig. 78. The Spectroscope. '"* ° f C ° lorS > vi ° let > indi S°> 

blue, green, yellow, 
orange, red. The actual spectroscope consists of a prism P 
(or a succession of several prisms) and two brass tubes. One, 
the " collimator" C, admits light to the prism. It has a 
narrow slit s, at one end, so that the light may enter as a 
thin line, parallel to the edge of the prism ; at the other end 
there is a lens, 0, to render the rays of light parallel. The 

282 






ASTRONOMIC INSTRUMENTS 

other brass tube is merely a little view telescope O'E, with 
which to examine the " spectrum" and to magnify it. Fre- 
quently it is better to substitute for the prism a glass plate 
on which a great number of very close parallel lines have 
been ruled with a dividing engine. Such a glass spreads 
light into a spectrum similar to the one obtained with a 
prism. Plate 15 shows a spectroscope attached to the eye- 
end of the big Lick telescope. Three prisms are used ; and 
the eye-piece of the view telescope is replaced by a plate- 
holder, so that stellar spectra may be photographed. 

If we send colored light into the spectroscope, the result 
is as follows : If the light is red, it goes to the red part of 
the spectrum, where it belongs, and we there see a bright red 
line, which is merely an image of the spectroscope slit. But 
if we send in white light, which is really a mixture of all kinds 
of colored light, the prism analyzes it : the yellow part goes 
to the yellow part of the spectrum, etc. We then see the 
colored " continuous spectrum," made up of an enormous 
number of slit-images side by side (see Plate 16, 1). But if 
the light comes from an incandescent gas, instead of a solid 
or liquid, we see no continuous spectrum, but a series of 
bright-colored lines, or images of the slit, variously located 
throughout the spectrum; and the combination of lines is 
different for every different gas. We can actually deter- 
mine the name of the incandescent gas from the positions of 
the lines. This is well shown in Plate 16, 3, 4, and 5, for the 
incandescent vapors of sodium, hydrogen, and potassium. 

But the most singular thing of all relates to the " absorp- 
tion" of light by gases; and this is one essential thing in 
astronomic spectroscopy. If a beam of white light is passed 
through a layer of gas or vapor before entering the spectro- 
scope, this vapor will sift out, and absorb, precisely those 

283 



ASTRONOMY 

light-rays or colors which the gas or vapor would itself 
emit if it were incandescent, and which would then appear as 
a bright-line spectrum in a spectroscope. 

From the foregoing principles we shall find it possible to 
study the chemical constitution, or the nature of the vapor 
existing in the sun and stars. And there is also another prin- 
ciple, due to Doppler, by means of which we can obtain in- 
formation of still another kind. It is a fact that the spectral 
lines may at times be shifted out of their proper positions as 
ordinarily seen in the spectrum. We are taught in the 
science of physics that red light has comparatively long 
light-waves ; violet light, the shortest waves. Now if a 
source of light, such as a star, happens to be increasing its 
distance from us at the time of observation, we shall receive 
fewer light-waves per second than would be the case if the 
distance were stationary. But if we receive fewer light- 
waves per second, they will seem to be longer waves, and 
therefore more like the waves from red light. The effect 
is to shift the observable lines toward the red end of the spec- 
trum. This shift can be measured with a micrometer, and 
from such measurements it is actually possible to deter- 
mine the velocity with which a star is approaching us or 
receding from us in space (cf. p. 245). 

The planet Venus has often been observed to test this 
principle ; and the photographed spectrum of Venus in 
Plate 16, 6, shows plainly the shift of the spectral lines. 
The middle spectrum belongs to Venus ; the two outer ones 
are artificial spectra produced in the observatory for compari- 
son. Microscopic measurement of this " spectrogram, " as it 
is called, enables us to compute that Venus was increasing 
its distance from the earth at the rate of eight miles per 
second when the spectrogram was made. From our knowl- 

284 . 





PLATE 1 6. Various Spectra. 



ASTRONOMIC INSTRUMENTS 

edge of the orbital motions of Venus and the earth, it is 
possible to check this result by an entirely independent method 
of calculation ; and thus no doubt remains as to the correct- 
ness of the Doppler principle. 

In addition to the spectroscope just described, astronomers 
also employ a " slitless " instrument. This is made by mount- 
ing one or more prisms outside the object-glass of a telescope, 
prisms large enough to cover the entire object-glass. With 
such an instrument it is possible to photograph on a single 
plate the spectra of all stars in the telescopic field of view, 
while the slit spectroscope will give only one star-spectrum 
at a time. It is therefore clear that the slitless instrument is 
best for statistical researches intended, for instance, to classify 
all spectra; but the slit instrument is more accurate, and 
is also the only form permitting the measurement of line 
shift in accordance with Doppler's principle. 



285 



CHAPTER XVI 

SUNSHINE 

In Chapter XIV we have considered at length the ancient 
problem of determining the distance of the sun from our 
earth: let us next attempt to describe the sun itself. 
Here we meet something distinctly modern in the venerable 
science ; for almost all knowledge we have of the sun is 
knowledge obtained during the last hundred years. The 
ancients knew little or nothing about it. 

Our subject falls readily into two parts : first, information 
obtained by the use of the spectroscope (p. 282) ; and 
secondly, investigations other than spectroscopic. Let us 
begin with the chemistry of the sun. We have already had 
an explanation (p. 283) of the manner in which gases absorb 
certain light-rays while passing through them, each gas ab- 
sorbing its own particular combination of such rays. This 
principle, applied to the sun, gives the following result. 
The body of the sun sends out white light which the spec- 
troscope would naturally split up into a continuous spec- 
trum (p. 283). But before this white light can pass through 
the outer gaseous layers of the sun, the absorption phenom- 
ena take place ; and so the regular solar spectrum appears 
in the spectroscope as continuous, but crossed by a vast 
number of black lines. These lines are simply the dark 
places where absorption has occurred. Thus, for instance, if 
there is iron vapor in the outer solar atmosphere, we shall 
see dark lines in the solar spectrum at exactly the points 

286 



SUNSHINE 

where we should see bright lines (p. 283) if we vaporized some 
iron in the laboratory, and examined the spectrum of its light. 
All these dark lines are merely bits of nothingness occupying 
the places where there should be images of the spectroscope 
slit, if absorption were absent. Such are the famous 
Fraunhofer dark lines, named from their discoverer. 

Plate 16, 2, shows a number of the principal ones, together 
with the letters by which spectroscopists designate them. 
The double D-line, for instance, arises from absorption due 
to sodium vapor, as may be seen from a comparison with 
the sodium vapor spectrum, 3. 

So much being premised, we can now explain the method 
of using the spectroscope to ascertain the sun's chemical com- 
position. It is merely necessary to attach a spectroscope to 
an ordinary telescope in such a way that the slit will be in 
the telescopic focal plane (p. 272). Arrangements must then 
be made, by means of a tiny reflecting prism attached to 
the slit, to throw into the view telescope the spectrum of 
any desired substance vaporized and heated to incandes- 
cence in the observatory, near the telescope. We then see 
the solar spectrum and the artificial spectrum side by side. 

Now this artificial spectrum is a bright-line spectrum 
(p. 283) : and if opposite each of its bright lines we find a dark 
Fraunhofer line in the solar spectrum, we have conclusive 
proof that the substance vaporized in the observatory is 
actually present as a gas in the outer atmosphere of the 
sun. Many terrestrial chemical elements have been thus 
found in the sun : the general conclusion is that earth and 
sun have a similar constitution, which was to be expected, 
if we accept any hypothesis postulating a common origin 
for the sun and earth. (Cf. nebular hypothesis, p. 235.) 

Next we must consider a very interesting phenomenon 

287 



ASTRONOMY 

called the Reversing Layer. We have seen that the Fraun- 
hofer dark lines are due to absorption in the outer gases of the 
sun. But these gases are themselves so hot as to be incan- 
descent. It is only because the inner sun is so much hotter 
and so much more luminous that we ordinarily see only the 
light from the inner sun and not from the outer gases. The 
latter are dark by comparison only. 

There is just one occasion when it is possible to observe 
the light from the outer gases separately and directly. 
This occurs during a total solar eclipse, when the moon hap- 
pens to pass accurately in line between the earth and the 
sun. On such an occasion, when the lunar globe, advancing 
in its orbit around the earth, has almost covered the sun, 
just before it is covered absolutely, there must be a moment 
when a tiny sickle of the outermost layer of the sun is alone 
visible. At that exciting moment, and at that moment only, 
can we look upon the outermost incandescent gases. 

But their light suffers no further absorption ; and so, like 
all incandescent gases, should give a spectrum consisting of 
bright lines only. And this is precisely what occurs. If we 
observe the advancing eclipse, just for an instant before 
totality, the continuous solar spectrum with its myriad 
black Fraunhofer lines is suddenly replaced by a bright- 
line spectrum. Each bright line corresponds accurately to 
one of the vanished dark lines, since the dark lines were 
caused by absorption due to the very gases that are now 
furnishing the bright-line spectrum. The critical instant 
over, the sun is covered totally, and the bright lines in turn 
disappear, too. This phenomenon is appropriately termed 
the Flash Spectrum. 

The surface we see when we turn a telescope upon the sun 
is called the Photosphere. It is not uniformly brilliant, 

288 




PLATE 17. The Sun. 



Photo by Fox. 



SUNSHINE 

but shows certain brighter " nodules, " and also especially 
bright points called "faculae," 1 which appear mostly at the 
edges of the sun and near the sunspots. They are perhaps 
suspended in the solar atmosphere at a high level, and owe 
their extra brightness to our observing them through a 
somewhat thinner layer of solar atmosphere. Probably 
everything we see when we examine the sun is " atmosphere.' ' 
Young aptly compares the state of affairs to a gas-burner, in 
which the heated particles of the mantle are far more lumi- 
nous than the flame of gas which heats them. 

Let us next enumerate some of the principal facts known 
about the sunspots (p. 17). We are on sure ground when 
we speak of their size ; for we can, as usual, measure their 
angular diameters, and then compute their linear dimensions 
from our knowledge of the sun's distance. At times they 
are 50,000 miles in diameter; and exceptionally large ones 
can even be seen without a telescope. But our knowledge is 
less certain when we attempt to explain their cause. They 
are to be regarded probably as solar atmospheric disturbances 
or storms. In that case we should expect them to shift their 
positions on the sun's surface, much as storm-centers move 
across our earth. And we find by observation that all spots 
have a common drift; and those near the solar equator also 
drift toward it, while those far from the equator drift toward 
the solar poles. This might be analogous to our phenomena 
of the trade-winds, especially as spots never occur near the 
solar poles, or exactly at the equator. 

So the real cause of the spots must be regarded as un- 
known. They may be eruptions from the interior; they 
may be gases rushing downward into hollows. But we 
cannot help thinking they are vast storms of some kind; 

1 Plate 17. 
U 289 



ASTRONOMY 

storms of which the materials are incandescent gases, moving 
with great velocities, and at enormously high temperatures. 

The duration of individual spots is not great, never more 
than 18 months ; and the central, apparently blackest part 
of the spots, called the " umbra," is not really dark, but 
appears so only though contrast with the much more lumi- 
nous surrounding solar material. They have also a periodic- 
ity, discovered in 1843 by Schwabe ; and this is perhaps the 
most interesting of the many unexplained observations of the 
sun. Schwabe found, by constant watching of the solar 
surface, that every eleven years there is a period of extra 
great spot frequency. This discovery owes its importance 
to the known fact that there exists also an eleven-year 
period in the frequency of terrestrial magnetic storms : 
and especially great sunspots are always accompanied with 
very strong magnetic disturbances and auroral displays on 
earth. This establishes the existence of some intimate 
magnetic relation between earth and sun; but it has not 
yet been possible to reach a satisfactory explanation of it. 
Nor have astronomers been able to make certain that any 
other terrestrial meteorological phenomena exhibit a real 
connection with the spots, though many efforts have been 
made to do so on account of the assistance such investigations 
might give in the matter of weather prediction. 

The accompanying Plate 18 is a photograph of a partic- 
ularly large sunspot which occurred July 17, 1905. It had 
an unusual and very brilliant " bridge" across the umbra. 

The next important question requiring consideration re- 
lates to the size of the sun. We have already determined its 
distance to be about 93 million miles. Observations very 
similar in principle to methods already explained for the 
moon and planets enable us to measure the sun's apparent 

290 



SUNSHINE 

angular diameter (cf. p. 203); and this we find to be 32' 4", 
on the average. In Fig. 79 we then have, as usual, a long, 
narrow triangle, of which the base is the sun's linear diameter 
AB. This we can calculate readily, because we know the 
sun's angular diameter, or the angle 32' 4" at the vertex 




To the Earth 
Fig. 79. Sun's Diameter. 

of the triangle, situated on the earth. The result comes 
out nearly nine hundred thousand miles for the sun's linear 
diameter. 

Comparing this with the known terrestrial diameter 
(p. 97), we find that the sun's diameter is approximately 110 
times that of the earth. 

To ascertain the sun's mass is a little more difficult than to 
find its diameter ; but it can be estimated by simple mathe- 
matical methods, 1 which show that it is about 330,000 
times the earth's mass. 

We can also calculate the force of gravity that must exist 
on the solar surface as compared with the gravitational at- 
traction existing on our earth. For the gravity force on the 
surface of a sphere is, by Newton's law, proportional to the 
mass of the sphere, divided by the square of its radius. If 
we then consider all solar quantities expressed in terms of the 
corresponding terrestrial quantities as units, we have : 

c , , r solar mass 330000 „ , . 

Solarforceof gravity = (sdar radiug) 2 = -^jt = 28, approximately. 

This means that an ordinary one-pound weight would 

1 Note 35, Appendix. 
291 



ASTRONOMY 

weigh 28 pounds, if transported to the sun's surface, and 
there weighed with an ordinary terrestrial spring-balance. 

The solar volume or bulk compares with that of the earth 
in the proportion of the cubes of their radii ; that is, as 1 to 
(HO) 3 . This makes the solar volume 1,300,000 times the 
earth's. And since density or specific gravity is proportional 
to mass divided by volume, it follows that the solar density, 
as compared with the earth's, is : 

solar mass 330000 

solar volume ~ 1300000 ~ °' 25 ' 
or only about J the earth's density. The latter, com- 
pared with water, is about 5.5 ; so the solar density is only 
about 1J times that of water. This means that a cubic foot 
of average solar material, transported to the earth's surface, 
would there weigh only about 1^ times as much as a cubic 
foot of water. 

The quantity of light and heat received by us from the sun 
is certainly enormous; and yet it cannot be more than a 
small fraction of the total quantity actually radiated into 
space. A most interesting question arises in connection 
with this matter : How does the sun maintain through the 
ages so gigantic an output of heat energy? What is the 
source of the sun's heat? Helmholtz has proposed a 
plausible possible cause, — the shrinkage of the sun's vast 
bulk under the influence of its own gravitational attraction. 
He computed that an annual shrinkage of only 300 feet 
in the solar diameter would be transformable into enough 
heat energy to keep radiation active as it now is. And it 
would require 8000 years for this diminution of size to 
reduce the sun's observable angular diameter by a single 
second of arc ; nor could any smaller diminution be discov- 
ered by observation with actual astronomic instruments. 

292 



SUNSHINE 

The fact that we have not observed a reduction of the sun's 
size is therefore no argument against the Helmholtz theory. 

Up to this point we have supposed the sun to consist of 
a highly heated interior of more or less unknown constitu- 
tion, surrounded by an atmosphere of incandescent gases 
which produce the Fraunhofer lines by absorption, and the 
bright-line flash spectrum during an eclipse. But we 
know much more than this. Beyond the photosphere 
and the reversing layer of gases is the Chromosphere, or 
color sphere, composed principally of great flaming masses of 
red hydrogen vapor. Sometimes great red jets burst out- 
ward to immense elevations from the solar surface. These 
are the Prominences (Plate 19) ; and while these various 
solar layers have different names, it must not be supposed 
that they are distinct. They intermingle, doubtless, at 
their boundaries, and melt into each other without sudden 
interruptions. 

The hydrogen prominences were first seen during a total 
solar eclipse, when the photosphere was completely covered 
by the moon. But just after the eclipse of 1868, Janssen and 
Lockyer for the first time succeeded in observing them with- 
out an eclipse. We cannot see them by merely looking at 
the sun with a telescope, and covering the central part of 
the solar disk at the telescopic focus, because the terrestrial 
atmosphere is strongly illuminated by the sun itself, and so 
the prominences become invisible by contrast. But if we 
bring the slit of a spectroscope tangent to the sun's disk at the 
focus of a telescope, and open the slit wide, the prominences 
become visible. 

For we then see two spectra superposed, one upon the 
other. The first is an ordinary continuous solar spectrum 
derived from the diffused sunlight in the terrestrial atmos- 

293 



ASTRONOMY 

phere, the second a bright-line spectrum from the incandes- 
cent hydrogen of the prominences. Now, if we employ in the 
spectroscope a number of prisms, instead of a single one, 
both these spectra will be spread out to a great length. 
The continuous spectrum will be thereby rendered dimmer, 
but the bright-line spectrum will have its lines separated 
further, without rendering them less brilliant. If we con- 
tinue thus increasing the " dispersion' 1 of the spectroscopic 
prisms, we shall finally diminish the luminosity of the at- 
mospheric continuous spectrum until it disappears practi- 
cally, and we see only the bright-line spectrum of the promi- 
nence. 

Now, as we know, these bright lines are ordinarily merely 
images of the slit. But if the slit has been opened wide 
enough to be wider than the angular diameter of the promi- 
nences, the bright lines become images of the prominences, 
instead of images of the slit. We have therefore merely to 
point the view telescope at a place in the spectrum where 
there is ordinarily a bright hydrogen line, and we shall see 
the prominence, if there happens to be one on the sun's 
edge at the point where we have placed the widely open 
slit tangent to the sun's image at the telescopic focus. 

In 1891 Hale invented an instrument called a spectrohelio- 
graph, with which the prominences may be photographed 
without an eclipse. Plate 19 was made with such an instru- 
ment. It utilizes the light of calcium gas, which, like hydro- 
gen, is plentiful in the prominences ; and can be made to 
give a fine bright line in the middle of the usual dark Fraun- 
hofer calcium line due to the photosphere and reversing 
layers. The spectrum is allowed to fall on a screen having a 
second narrow slit corresponding accurately to the bright 
calcium line from the prominence. Through this slit 

294 



SUNSHINE 

the calcium light which originated in the prominence passes 
to a photographic plate, so that the plate receives prominence 
light only. Now, by mechanical means, the original slit 
of the spectroscope is moved across the solar image at the 
telescopic focus ; and the second slit in the screen is moved 
in unison. The result is to build up a picture of the sun on 
the photographic plate with light from the outer solar layer 
only, and thus to secure a photograph of the prominences. 

Still another extraordinary solar phenomenon has been 
discovered during total eclipses. This is the Corona, which 
bursts into view when the sun is completely concealed by 
the moon, and appears as a faint luminous ring, of more or 
less irregular shape, around the sun. We know that it 
belongs to the sun, because its spectrum is that of an in- 
candescent gas, not a continuous solar spectrum, such as 
it would be if we had here to do merely with solar light 
reflected in some way from the moon ; and also because the 
coronal form always appears the same at the same moment, 
even when seen from observatories widely separated on the 
earth. Beyond this little is really known for certain as to 
the corona. 

Plate 20 is a photograph of the corona during a total 
eclipse, the sun's disk being entirely covered by the inter- 
posed moon. The form of the streamers indicates that the 
phenomenon may be electric or magnetic. A large promi- 
nence appears near the lower part of the plate, jutting out 
from the obscured solar disk. The height of this prominence 
is estimated by comparing it with the photographed diameter 
of the sun : this would make the prominence about 40,000 
miles high. 

The question of solar axial rotation can be examined best 
by studying the apparent motions of the spots. (Cf. the 

295 



ASTRONOMY 

case of the planets, p. 202.) It is found that they always 
seem to cross the solar disk from east to west; and when 
one of them makes a complete rotation, disappearing at 
the western, and later re-appearing at the eastern edge of 
the sun, the whole revolution takes about 27| days. But 
this is not the true period of solar axial rotation : the above 
observed period, and the true one, are related by an equation 
analogous to the equation connecting the synodic and sidereal 
periods of the planets (p. 209), since we must correct the 
observed period for the effect of the earth's orbital motion 
around the sun during the time occupied by the latter in 
turning on its axis. 1 This correction makes the true axial 
rotation period of the sun about 25 J days. 

These sunspot motions not only tell us the period of the 
sun's axial rotation; they also enable us to ascertain the 
direction in space of the rotation axis (cf. p. 203). The 
apparent paths of the spots are generally curved ; but on 
June 3 and December 5 they appear quite 
straight. Referring to Fig. 80, we see that this 
straightness determines on these dates the axial 
Fig. so. Sun's or polar points A and B on the sun's edge, and 
also the angle by which the solar rotation axis 
is inclined to the plane of the ecliptic. The angle is about 
83°. 

1 The planetary equation here takes the form : 

1 1 1 




true period of rotation length of year observed period of rotation 



296 



CHAPTER XVII 

ECLIPSES 

Eclipses are occasional phenomena; they are usually- 
defined as temporary obscurations of the sun or moon, 
either wholly or in part. We have seen that these two 
bodies are visible from two very different causes : the sun is 
self-luminous, — it is 

. ., , , .. i S \ Solar Eclipse 

visible because it sends 




us its own light ; the V " / Moon 

moon is merely ren- 
dered visible when il- 
luminated by the sun. /^~^\ Lunar Eclipse ^-^ 

Therefore a solar I &un J \^y M ^ n 

echpse can occur only FlG 81 Eclipses 

if the moon inter- 
poses between us and the sun, thereby preventing our see- 
ing it; but a lunar eclipse happens when the earth passes 
between the moon and sun, so that solar light cannot reach 
the moon, and render it visible. Thus there is not neces- 
sarily any actual obstruction in the way of our seeing the 
eclipsed moon ; it is invisible merely because it is dark for 
the time being. Figure 81 makes all this plain. 

Hipparchus was the first to explain eclipses, and the 
method of making a fairly good approximate prediction of 
their occurrence. Figure 81 shows that if the orbits of the 
earth and moon were both situated in a single plane surface 
(here represented by the plane of the paper) there must 
result one eclipse of the sun and one of the moon during 

297 



ASTRONOMY 

each revolution of the moon in its orbit around the 
earth. This simple state of affairs is modified and com- 
plicated by the fact that the lunar orbital plane is actually 
inclined about 5° to the ecliptic plane, in which the earth's 

orbit is situated (p. 
jfooniSi^ — 160). Figure 82 is 




supposed to represent 

& Ecliptic 

Fig. 82. Moon's Circle. a PO^lOn of the Sur- 

face of the celestial 
sphere. NS is part of the ecliptic circle, in which the sun is 
always seen ; and NM is part of the great circle cut out on 
the sphere by the plane of the moon's orbit, in which great 
circle the moon is seen, for the same reason that the sun is 
always seen in the ecliptic (p. 160). N is the point of inter- 
section of these two great circles on the celestial sphere, the 
angle between them being 5° only. The point A r is called 
the "node" of the lunar orbit; and there is another similar 
node on the opposite side of the sky, because any pair of 
great circles must necessarily intersect at two opposite 
points on the sphere. 

We have already seen that the moon moves around the 
earth, and therefore appears to travel around the sky among 
the stars, at the rate of about 13° per day, so that it overtakes 
and passes the sun once in each synodic period or lunar 
"month" (p. 161). When it thus passes the sun, the two 
bodies are said to be in conjunction (cf. p. 209). If this 
conjunction happens to occur exactly at the nodal point N, 
then sun, moon, and earth will lie in a single straight line, 
and a "central" eclipse will occur. Furthermore, it will 
be an eclipse of the sun; for if the sun and moon appear 
projected at a single point of the sky, they must both lie 
on the same side of the earth (Fig. 81, solar eclipse). 

298 



ECLIPSES 

But an eclipse can also happen when the sun and moon 
are in opposition, or 180° apart, as seen projected on the 
sky. If such an opposition takes place when the moon is 
exactly in the node N, and the sun in the other, or opposite, 
node, we once more have sun, earth, and moon in a single 
straight line, and a central eclipse takes place. Only, in 
this case, the earth is between the sun and moon (Fig. 81, 
lunar eclipse), and the central eclipse is a lunar eclipse. 

The problems connected with eclipse prediction would pre- 
sent but little of interest beyond the above, if the node N 
always remained at the same point of the ecliptic. But this 
node is constantly moving along the ecliptic, — a phenomenon 
somewhat analogous to precession of the equinoxes (p. 126) 
in the case of the earth. Only, the lunar node, unlike the 
equinoxes of the terrestrial orbit, moves quite rapidly, 
making a circuit of the entire ecliptic once in about 19 years. 
It is this phenomenon that complicates the eclipse problem 
and makes it interesting. 

Up to this point we have supposed eclipses to occur at 
the exact nodes only ; and this would be the case if the sun 
and moon appeared to us on the sky as mere 
mathematical points, like the fixed stars. 
But both sun and moon have quite large 
disks, as we see them projected on the sky. 
Each disk has a diameter of about half a 
degree of arc. Consequently (Fig. 83), when Fig. 83. Contact 
a conjunction occurs, these two disks may 
just touch if their centers are half a degree apart, provided 
we suppose the terrestrial observer located at the earth's 
center. 

But an observer on the surface of the earth, as at the point 
in Fig. 84, will see the lunar and solar disks in contact when 

299 




ASTRONOMY 

the moon is at Mi; while to an observer at the earth's 
center c, there would be no contact until the moon had 
advanced to M 2 . It is clear from Fig. 83 that the angle 
ScM 2 is |°. The angle ScMi (from the center of the sun 

to the center of the moon, as 

■^— -i qm; q seen from the center of the 

f s' — A Q -^ - T ) earth) is 1^°, approximately, 1 

\^S Earth thus enlarging greatly the 

Fig. 84. Observe^the Surface of the possibility Q f an eclipse being 

actually visible from some 
point or other on the earth's surface. 

Now referring again to Fig. 82, we see that if the conjunc- 
tion occurs when the centers of the sun and moon are at 
the points S and M, just far enough from the node N to 
make these two points 1|° apart, the two disks will just 
touch, and we shall barely escape the occurrence of an eclipse, 
visible from some point of the earth's surface. If M and S 
happen to be a little nearer the node, the two disks will 
overlap, and there must be at least a partial solar eclipse. 

Knowing the angle at N to be 5°, there is no difficulty in 
calculating how great must be the angular distances NM and 
NS, to make the distance SM just 1J°. This distance NM 
is thus found to be about 17° ; so that when conjunction 
occurs within about 17° of the node, there is an eclipse. 
But this number 17° may vary all the way from 15° to 19° 
in different years, largely on account of small periodic 
changes of the angle N, between the two orbital planes. 
These changes are of course due to orbital perturbations 
(p. 206). The number 17° is called the "solar eclipse limit." 

As we have seen, these eclipses at conjunction are solar 
eclipses ; but corresponding eclipse limits exist also in the 

1 Note 36, Appendix. 
300 



ECLIPSES 

case of oppositions of the sun and moon, when lunar eclipses 
occur. Referring again to Fig. 81, we see that a solar eclipse 
is possible, only if the disks of the sun and moon, as projected 
on the sky, actually overlap. But Fig. 85 makes plain that 
a lunar eclipse will happen if the moon enters or touches the 
shadow cast into space by the earth. But the apparent 
angular diameter of this shadow, as seen from the earth, 




Fig. 85. Lunar Eclipse. 

and at the distance of the moon from the earth, is larger 
than the moon's angular diameter. The result is that a 
lunar eclipse takes place if the center of the moon and the 
center of the shadow are separated by an angular distance 
of less than about 1°, as seen from the earth. We can, of 
course, again calculate how far the sun must be from the 
node, at the time of opposition, to make this angular distance 
less than 1°. We thus find the lunar eclipse limit of about 
11°, with a variation between 10° and 12°, in round numbers. 
Since conjunction always happens at new-moon, and op- 
position at full-moon, it follows from the foregoing simple 
considerations that there will be a solar eclipse at the time 
of new-moon, if the sun is within about 17° of the node; 
and there will be a lunar eclipse at the time of full-moon if 
it is within about 11° of the node. Two other simple conclu- 
sions follow at once : (1) Since the sun appears to move about 
1° daily in the ecliptic, there will be a solar eclipse if the date 
of new-moon falls within 17 days of the date when the sun 
appears in the node. And there will be a lunar eclipse if 
the date of full-moon falls within 11 days of the same date, 
all in round numbers. (2) The solar eclipse limit being 

301 



ASTRONOMY 

the larger, solar eclipses must be more frequent, on the whole, 
than lunar eclipses. 

Hipparchus was the first to explain correctly the seeming 
paradox that lunar eclipses are seen much more often than 
solar eclipses, although the latter occur more frequently. 
The reason is perfectly simple. When the earth interposes 
between the sun and moon, and the moon thus enters the 
earth's shadow, it becomes dark at once, because it gives no 
light of its own. Consequently, any one on the earth who 
should be able to see the moon will fail to see it on account 
of the eclipse. But at any given instant, the moon should 
be visible from half the earth's surface ; therefore, if there 
is a lunar eclipse, at least half the earth's inhabitants will 
see it. 

But in the case of a solar eclipse, the sun is not made dark. 
The sun's light is actually cut off by the interposed moon ; 
and it is never at any one time thus cut off from observers 

, M living in more than a small 
part of the earth's surface. In 
Fig. 86, an observer on the 
earth at A will see the moon 
projected in his zenith, just as it 
would be seen by an observer 
at the earth's center C. But 

Fig. 86. Solar Eclipse. 

an observer at B will see the 
moon in the direction BM , instead of AM. This will pro- 
ject it in the sky for the two observers at points whose 
angular distance apart is equal to the angle BMC. Now 
this angle may be as great as 1°, approximately ; so that the 
disks of the moon and sun might easily overlap for an 
observer at A, but not for an observer at B (cf. Fig. 84, 
p. 300). In other words, solar eclipses are by no means 

302 




ECLIPSES 

visible throughout an entire hemisphere of the earth, like 
lunar eclipses. In fact, the distances of the sun and moon 
from the earth are such that any total solar eclipse can 
be seen from a very narrow strip of the earth's surface 
only : not more than 70 miles wide at the most, and ex- 
tending through a distance of much less than a hemisphere. 
Plate 20 (p. 295) is a photograph of a total solar eclipse. 

Having thus outlined the general nature of eclipses and 
their causes, we shall next describe certain special phenomena 
which are of sufficient interest to be mentioned. The earth's 
shadow, into which the moon enters when eclipsed, is not 
uniformly dark throughout. It is, in fact, made up of two 
parts, — the ' ' umbra, ' ' or shadow proper, and the ' ' penumbra, ' ' 
or partial shadow. Figure 87 shows how the penumbra is 
formed. There are two penumbral regions, as it were, and 
one umbral region. The latter is a central cone which re- 




Fig. 87. The Penumbra. 

ceives light from no part of the sun. The two penumbral 
regions receive light from part of the sun, while the rest of 
space behind the earth receives light from the entire solar 
surface. It is evident that the darkness of the penumbra 
will increase gradually from its outer edges to the boundary 
lines, where it gives the black umbra. The moon, when 
about to be eclipsed, will therefore enter the penumbra 

303 




ASTRONOMY 

first, and be partially darkened; and the darkening will 
increase gradually until it becomes practically complete, as 
the moon enters the umbra. The same gradual phenomena 
will be repeated in the inverse order towards the end of the 
eclipse. 

In the case of solar eclipses (Fig. 88), if we consider the 
interposed moon as casting a shadow on the earth, the eclipse 
will be total where the true shadow cone cuts the earth, and 
partial where the penumbral regions meet the terrestrial 
surface. 

Owing to the ellipticity of the moon's orbit, and consequent 
variation in the distance between the earth and moon, it 

sometimes happens that the 
true shadow cone of Fig. 88 
does not quite reach the 

Fig. 88. Solar Eclipse. earth « In SUch a CaSe > in 

that part of the terrestrial 
surface for which the eclipse is central, the sun will appear 
as a luminous "annulus," or ring, with the central part dark. 
Such eclipses are called Annular eclipses, and occur, of 
course, for the sun only. A total solar eclipse can never last 
longer than eight minutes at any one place on the earth, 
but totality in the case of the moon may last a couple of 
hours. 

There exists a peculiar periodicity in the recurrence of 
eclipses called the Saros. It was discovered by the Chal- 
dseans, who found, by actual observation, and comparison 
with ancient records in their possession, that after the lapse 
of a period of 6585 days after an eclipse, the phenomenon 
will be repeated; and eclipse occurrences can thus be pre- 
dicted easily. The explanation is as follows : 

The reader will remember that in the case of the earth's 

304 



approximately 



ECLIPSES 

motion around the sun we found the tropical year to be 
shorter than the sidereal year by about twenty minutes, 
on account of precession of the equinoxes (p. 126). 

In a similar way, on account of the motion of the moon's 
nodes, the time required by the moon to travel in its orbit 
from one node back to the same node again is shorter than 
the lunar sidereal period (p. 161). This nodal period is 
called the Draconitic period ; it is three hours shorter than 
the sidereal period, and two days seven hours shorter than 
the synodic period. So we have : x 

Sidereal period = 27 d 8 h 
Draconitic period = 27 5 
Synodic period =29 12 

An inspection of these figures shows that 223 synodic 
periods equal 242 draconitic periods, very nearly; and 
either includes 6585 days. But successive full-moons, or 
successive new-moons, follow each other at intervals of 
one synodic month, because the synodic month is the in- 
terval between two successive overtakings of the sun by the 
moon, in their respective apparent motions around the 
celestial sphere. And successive passages through either 
node succeed each other at intervals of one draconitic period. 
Therefore, any period of days like the Saros, containing 
exactly a definite number of synodic periods and also a 
definite number of draconitic periods, — after the lapse of 
such a period of days, both the lunar phase and the node 
passage must both repeat. Therefore, if there was an 
eclipse at the first new or full moon of the Saros period, there 
must also be an eclipse at the first new or full moon of the 
succeeding Saros period. 

1 Note 37, Appendix. 
x 305 



ASTRONOMY 

In the light of the above explanation of eclipses, it may 
be possible to make somewhat clearer the allied phenome- 
non called a transit of Venus (p. 268). Such transits also 
occur in the case of Mercury, but they are then o- lesser in- 
terest. The planetary nodes do not move around the 
ecliptic rapidly, like the lunar nodes; they remain almost 
stationary at a definite point. The sun, in its apparent 
motion around the ecliptic, reaches the nodal points of 
Venus on June 5 and December" 7 ; so that transits of that 
planet happen (if at all) within a day or two of these dates. 
To ascertain the interval between successive transits, we 
note that : 

5 synodic periods of Venus = 8 years, nearly ; 
152 synodic periods of Venus = 243 years, very nearly. 

Since conjunctions of Venus occur at intervals of one sy- 
nodic period, any given transit may be followed by another 
at the same node eight years later. But there could not be 
a third transit sixteen years later; the eight-year period is 
not exact enough for that. We should then have to wait 
for the 243-year period to become effective. But at the other 
node, a transit, or an eight-year pair of transits, may happen 
after half the 243-year cycle has elapsed. 



306 



CHAPTER XVIII 

COMETS 

The comets, stellce cometce, or stars with hair, must next 
receive our attention. These bodies usually move in very 
elongated orbits, with the sun at one focus. They often 
come as mere occasional visitors to the solar system, are 
seen during a short period only, while they are traversing 
that part of their orbit which is near the sun and therefore 
also near the terrestrial orbit. Occasionally comets have 
been as luminous as the brightest planet (Venus) ; have 
sometimes been seen in daylight; and very often have 
a long appendage streaming out from the head, — the 
comet's "tail." 

It is easy to enumerate the chief known facts concerning 
the comet's physical appearance. The head usually con- 
sists of a "coma," or hazy nebula, containing a " nucleus," 
or central condensation. Attached to it is the tail, or, as 
it was sometimes formerly called, the "beard." The coma 
is the part that generally becomes visible first, as the body 
begins to approach the solar system. The nucleus forms 
later, or at least becomes visible later. The tail, strange 
to say, is always directed away from the sun; so that 
when the comet is receding from the sun, after passing the 
perihelion point (p. 120) of its orbit, it pushes its tail out 
ahead of it. But many comets were not discovered until 
after perihelion. It was then that the astronomers of old 
used the name "beard" for the tail. 

Comets are big; their volume is sometimes incredibly 

307 



ASTRONOMY 

large. The heads run up to a million miles in diameter, 
and the tails may be ten million miles long, or even much 
longer. But they have little mass, as is evidenced by the 
total absence of gravitational perturbations (p. 206) in the 
motions of the earth and Venus, even when a big comet 
passes very near these planets. Owing to this vastness of 
bulk and extremely small mass, comets have, of course, 
a very low density, especially in their tails. Moreover, 
stars have at times been seen through the comets, even 
through their heads. 

In view of this extreme lack of mass, it may seem strange 
that modern science is compelled to admit the possibility, 
at least, of danger resulting from collision between the 
earth and a comet. If the comet ary particles are infinitesi- 
mally small, no injury would follow ; but if the particles are 
rocks weighing tons, they might cause considerable local 
damage at the point of collision on the earth. 

But, on the whole, the most plausible theory is to suppose 
these bodies to be composed of tiny particles traveling 
together in swarms, and separated by distances many 
times greater than the diameter of the particles. And the 
particles may be surrounded by atmospheres of incan- 
descent gases; for we know that comets are partly self- 
luminous, although they send us also a certain amount of 
reflected sunlight, like the planets. This is made certain 
by observations of their spectra, which generally show the 
existence of hydrocarbon gas in a luminous state, as well as 
a dim continuous spectrum containing Fraunhofer (p. 287) 
lines, — the sure indication of solar light. 

We have a good theory to account for the repulsive forces 
that must come from the sun, so as to make the cometary tail 
always point away from that body. The researches of 

308 



COMETS 

Clerk-Maxwell (the same who proved mathematically 
the satellite construction of Saturn's ring, p. 245) have 
brought out the fact that light-rays exert a slight physical 
pressure upon any object they reach. This pressure is 
theoretically proportional to the area illuminated. 

Now if we imagine a small spherical cometary particle, 
and suppose its radius to diminish, the light pressure will 
diminish in proportion to the square of the radius, because 
the area of the circular cross-section illuminated diminishes 
in that proportion. But the volume and mass of the particle 
will diminish as the cube of the radius. Therefore the mass 
diminishes more rapidly than the area. But the solar gravi- 
tational attraction is proportional to the mass ; consequently, 
the solar attraction diminishes more rapidly than the light 
pressure, as the particle grows smaller. We have therefore 
merely to imagine the particle small enough, and the light 
pressure will balance the attraction. Still smaller particles 
will actually be repelled. 

If we now suppose the tail composed of particles smaller 
than those in the head, everything is explained : the head 
attracted, the tail repelled, by the sun. Probably the 
comet has no tail until it approaches the sun, when the light 
pressure sifts out the small particles; repells them in an 
increasing degree as the comet comes near perihelion ; and 
thus makes the tail "grow." Perhaps the tail particles 
never rejoin the head, but are left scattered throughout the 
length of the cometary orbit. If we finally suppose both 
gravitational attraction and light repulsion to be exerted 
on the several particles, both by the sun and the comet's 
head, we have a combination of forces sufficiently flexible 
to account for the most complicated observed forms of 
cometary tails. (See Plate 21, p. 307.) 

309 



ASTRONOMY 

The number of comets is very great ; while only some four 
hundred were observed before 1610, when Galileo first 
used the telescope, at least four hundred more have been 
found in the succeeding three centuries. Of course this 
greater abundance of discovery in modern times has been 
brought about by the possibility of recording comets too 
faint to be seen by the unaided eye. Only thirteen naked- 
tye comets belong to the nineteenth century. 

The method used in discovering comets is interesting. 
This work is carried on by specialists ; except as a result of 
unusual chance, one can expect to find new comets only after 
a number of years' severe study of the heavens. The usual 
process is to " sweep' ' the sky with a telescope of moderate 
size and low magnifying power. Any hazy object may 
be a comet; for when they are distant and dim, comets 
always look more or less like small nebulae. The only 
sure test to distinguish them is to watch for an hour or 
two, and ascertain whether there is motion relative to the 
fixed stars. If there is motion, a comet has been found. 
But this motion test occupies much time ; and at this point 
the experience of years is of value. For the comet hunter 
learns at last to know all the tiny configurations of stars 
seen in the telescope ; he knows the telescopic constellations 
at sight, as well as most astronomers know Orion and the 
Great Bear. It is said that Olbers, for instance, could tell 
the approximate right-ascension and declination of the 
point on the sky toward which his telescope was directed, 
by simply looking through the eye-piece, and noting the 
diagram of stars appearing in the field of view. 

And the comet hunter must know all the little nebulae too, 
as well as their positions on the sky relative to the sur- 
rounding small stars. When he sweeps a faint nebulous ob- 

310 




PLATE 22. Halley's Comet 



Photo by Curtis. 



COMETS 

ject into view, he does not ordinarily need to delay his work 
by testing for motion; he recognizes the object at once, if 
it is one of the known small nebulae. 

Comets are usually named after their discoverer, though 
a few bear the name of some person who has explained their 
motions or peculiarities by a special investigation. Thus 
the famous comet of Halley, a photograph of which is shown 
in the accompanying Plate 22, was the first comet for which 
future returns to the solar system were predicted by means 
of orbital calculations. These were made by Halley (cf. 
p. 269) ; and his name was accordingly assigned to this comet. 

Small telescopic comets are at first designated by the 
year of discovery and a letter ; as, 1899 a, etc. Later, when 
orbits have been computed, they take the number of the 
year in which their closest approach to the sun, or perihelion, 
occurs ; and a number in addition to show the order of 
cometary perihelion passages during that year. Thus 
Donati's great comet of 1858 was 1858 /, or the sixth comet 
discovered in 1858; and later it became 1858 VI, or the 
one whose perihelion passage was the sixth perihelion passage 
in 1858. 

The period of visibility does not usually last more than a 
few months, although its average duration has been length- 
ened considerably in recent years, because modern giant tele- 
scopes can observe the comets long after their orbital motion 
has carried them quite beyond the range of ordinary glasses. 

Concerning these orbital motions, the mists of antiquity 
certainly enshroud some very singular notions. There was 
a time when comets were regarded as material thrown out 
from the earth, possibly through volcanoes. It was not 
until ''bH that Tycho Brahe for the first time proved from 
actual measurements that the great comet of that year was 

311 



ASTRONOMY 



surely farther from the earth than is the moon. Kepler 
thought comets are alive. Hooke, in 1675, a century after 
Tycho Brahe, suggested that comet orbits might be para- 
bolic : a very few years later, Newton showed that they 
are " conic sections/' and Halley calculated actual orbits 
for all the comets observed up to that time. 

There are three kinds of conic sections, — the ellipse, para- 
bola, and hyperbola ; and it is easy to draw a figure illustrat- 
ing these three curves as comet orbits. Parabolic orbits 
are four times as frequent as elliptic orbits : hyperbolas are 




Fig. 89. Forms of Comet Orbits. 

very rare, and it is not absolutely certain that any such orbits 
really exist. Figure 89 exhibits the three orbital forms, 
together with the comparatively tiny circular terrestrial 
orbit, and the sun-dot at its center. We see especially 
how nearly alike all three kinds of comet orbits are, while 

312 




COMETS 

the comet is near the earth's path; and, after all, it is only- 
then that we can see a comet, and observe its position, 
as projected on the sky. To construct a comet orbit from 
observations is often as difficult as trying to draw a circle 
of large radius through three points very close together. 
Thus, in Fig. 90, it is 
easy to draw the circle ^ 
A accurately, through 
the three points Pi, P 2 , 
P 3 . But if we were 
asked to draw a circle 
through P/, P 2 ', P 3 ', 
we might not be able FlG " m Comet0rbits - 

to decide between the circle B and the circle C. 

Now the ellipse is a closed curve; the others are open 
curves : therefore only elliptic orbits will produce so-called 
"periodic" comets, with future returns to the solar system. 
The parabolic comets visit us once, and never return. But 
since parabolic orbits closely resemble extremely elongated 
ellipses, it is not always possible to make certain whether 
any given object is periodic or not. But it is, after all, really 
immaterial whether a given comet orbit is truly parabolic, or 
elliptic, with a period of several hundred thousand years. 

The exact details of any comet orbit are defined by means 
of elements exactly analogous to the elements of a planet's 
orbit (p. 200). 

So much being premised about these orbits, we can now 
consider one of the most interesting things known about 
comets, — their " families." For there are in existence most 
curious kinships between various groups of comets. Coming 
back to Kepler's amusing notion that they are alive, we 
must expect to find close relationships among them, and 

313 



ASTRONOMY 

also some that are merely distant cousins, as it were. The 
most remarkable " blood-relations " are the great comets of 
1668, 1843, 1882, and 1887. They must be brother-comets, 
for they all pursue practically the same orbit, though travel- 
ing in different parts of it. They approached the solar 
system from the direction of the bright star Sirius, and left 
again in nearly the same direction, in a parabolic orbit. 

On the other hand, there are the comet-families belonging 
to the great planets, especially Jupiter. Here all the 
comets of a family have the peculiarity that the points of 
their orbits farthest from the sun, the aphelion points, 
all lie near the orbit of Jupiter. In other words, they recede 
from the sun just far enough to reach Jupiter's orbit. If 
Jupiter happens to be in the neighborhood when they get 
out there, he must exert a powerful gravitational attraction 
upon them. It is supposed that on their first arrival, per- 
haps in parabolic orbits, this attraction pulled them around 
into ellipses, having their node and aphelion point near the 
place where this disturbing pull took place. This is the well- 
known capture theory of comets, due to Laplace. 

So we see that the comets did not originally belong to our 
solar system; they come to us from outer space, possibly 
from among the fixed stars, possibly from some nearer region. 
If they come from interstellar spaces, we should, on the 
whole, expect to find a preponderance of orbits having their 
aphelion points lying in the direction of that point on the 
sky toward which the solar system's own motion in space is 
tending. For the solar system, as a whole, is drifting through 
cosmic space, as will be explained in a later chapter. But 
we have only slight indications of such a clustering of aphelion 
points : our whole theory as to comet origins is till hazy, 
very hazy. 

314 



CHAPTER XIX 

METEORS AND AEROLITES 

The consideration of comets leads us directly to the 
closely related subject of meteors or " shooting stars." These 
look like stars falling from the sky ; actually, they are small 
particles of matter traveling in space, and passing through the 
earth's atmosphere. They give a bright light, and usually 
leave a long visible trail behind them. Sometimes they do 
not appear merely as isolated bodies ; but regular showers 
occur, with the bright intermittent trails almost covering 
the sky, or a portion of it, for a con- 
siderable time. When this happens, it 
has been found that all the meteor 
trails are directed away from some 
single point on the sky. This point 
is called the Radiant ; and it has the 
peculiarity that the trails are always f \ 

short in its vicinity. Figure 91 exhib- 
its this state of affairs. The point R 
on the celestial sphere is the radiant. Fig. 91. Radiant of Meteor 

Shower. 

All the trails are directed away from 

it, as shown by the arrows ; and the longer trails originate 

at points farther from the radiant than do the short trails. 

Figure 92 shows that the whole appearance is due simply 
to perspective. The meteors move in parallel lines to meet 
the earth. Suppose the observer to be on the surface of 
the earth, at 0, and two meteors moving along parallel lines, 

315 



ASTRONOMY 



* 



B 



'-^ N 



1/ 



>Ci 



•^•^g/" 



1/ 



Surface ofihe Earth 
Fig. 92. Radiant of Meteor Shower. 



such as AiBi and A 2 B 2 . To the observer at the meteors 
will seem to move along the lines AiCi and A 2 C 2 . And all 
the meteors will seem to move along lines that will appear 
to radiate out from a single point R, where AiCi and A 2 C 2 
intersect, if produced backwards from A± and A 2 . And 

^ this point will appear 

in the sky near the 
meteors that seem to 
have short trails. 

Each meteor shower 
can be distinguished 
from all others ; not by 
a difference in the ap- 
pearance of its constituent meteors, but by the position on 
the sky of its radiant. Thus the shower called the. Leonids, 
the greatest of all the showers, has its radiant in the con- 
stellation Leo (Fig. 20, p. 62). These meteors occur always 
about November 12, and have been found especially numerous 
at intervals of 33 years. 

The cause of this fixity in the dates and recurrences of 
Individual showers is quite simple. Each shower travels 
in a definite orbit around the sun, just like a periodic comet 
(p. 313). This orbit somewhere intersects the orbit of the 
earth; or, at least, passes very near it. But the earth 
must reach that point of intersection on the same date each 
year. Therefore the shower must occur on that particular 
date, if it occurs at all. And it will occur if the meteors hap- 
pen to be at the intersection point of the two orbits on the 
date when the earth also reaches that point. In the case of 
the November Leonids this happens only once in 33 years ; 
but the Perseids, or August meteors, are ready for us every 
year. We conclude, of course, that the Perseids are spread 

316 



METEORS AND AEROLITES 



out all along their orbit, so that we meet some of them when- 
ever we strike the orbit. But the Leonids must be concen- 
trated in a certain region in their orbit : this region comes 
around to the point of intersection in the proper way only 
once in 33 years. 

A very interesting fact about the meteors is that we or- 
dinarily observe more of them per hour just before sunrise 
than we do just after 

. ( ij^jc Circle 

sunset. The reason is 
shown in Fig. 93. E\ is 
the position of the earth 
in its orbit at about six 
in the morning, just be- 
fore sunrise. The sun is 
seen projected on the 
ecliptic at Si, which is 
therefore near the east 
point of the horizon. 
The earth's orbital mo- 
tion is for the moment 
directed towards the 
point P, 90° west of S, 
where the sun appears on the ecliptic. The earth's orbital 
motion takes place in the direction of the curved arrow, so 
that at six in the morning we are on the front of the earth 
in respect to its orbital motion ; we advance to meet the 
meteors. But on the opposite side of the earth it is at the 
same instant 6 o'clock in the evening. There they see only 
such meteors as overtake the earth, while on the front side 
we see them all. After the lapse of twelve hours, the earth 
has made a half-turn on its rotation axis ; conditions are 
reversed ; and we then have our clocks at six in the evening. 

317 




Fig. 93. 



30°Wesiof+heSun 
Meteors at Sunrise and Sunset. 



ASTRONOMY 

We are then, in our turn, on the back of the earth with 
respect to the direction of its orbital motion. 

Meteors are never seen until they enter the atmosphere of 
our earth, but their heat and light are not due to atmos- 
pheric friction in the ordinary sense. Sometimes it is said, 
erroneously, that they are "set on fire" in the same way as 
a friction match is lighted by being rubbed on a rough 
surface. Their fight is really caused by the compression of 
air in advance of the moving meteor. It is harder for the 
meteor to move against the compressed air ; this retards its 
motion, and the motion energy is transformed into heat energy 
(p. 2). Doubtless both the meteor and the air are heated. 

When a moving body is retarded by the resistance of 
an atmosphere, the heat engendered is proportional to the 
square of the velocity of motion. At the usual meteoric ve- 
locity, the temperature produced is probably equivalent to 
several thousand degrees Fahrenheit; and this, of course, 
will melt almost any substance. The rapid motion through 
the air then tears off particles of heated incandescent matter 
from the melted meteoric surface ; and these particles are left 
behind to form the tail or trail of the meteor. It is not 
known just why it sometimes remains visible for many 
minutes. Plate 23 is a photographic reproduction of a 
meteor trail, showing two remarkable variations of brilliancy. 
The negative also contains a couple of interesting nebulae of 
irregular form. 

It is altogether probable that the meteors, and especially 
the meteoric showers, are nothing else but fragments of dis- 
integrated comets. As soon as periodicity in the recurrence 
of showers was recognized, and it thus became plain that the 
meteors travel in definite orbits, it was but a short step to 
compare those orbits with known cometary paths. And 

318 




PLATE 23. Meteor Trail. 



Photo by Barnard. 



METEORS AND AEROLITES 



soon after the great Leonid shower of 1866, Schiaparelli 
showed that the Perseids, or August meteors, are in the same 
orbit as a comet discovered by Tut tie in 1862. And it 
was not long before the Leonids were similarly identified 
with the comet of 1866, discovered by Temple. 

At least eight different meteor showers are now known to 
be connected with comets. The conclusion is possible that 
the comet is itself but a condensed place in the meteoric 
procession; the meteors themselves the disintegrated part 
of the material involved in the whole transaction. Certain 
it is that at least one comet (Biela's) has actually been seen 
to break up. It was discovered in 1826, re-appeared in 
1846 according to prediction, and was seen to break in two 
during this period of visibility. In 1852 it was seen again, 
the two parts being now widely separated ; and it has never 
been visible since. In its orbit, however, moves one of the 
big meteor swarms. 

It is very important to determine as accurately as possible 
the height of meteors above the earth's surface. For this 
is about the only direct 
method we have to ascer- 
tain observationally the 
extent of the terrestrial 
atmosphere. Since the 
meteor becomes visible 
only when it penetrates 
the earth's envelope of 
air, we shall know something about the height of the air 
if we can measure the height of the meteor. The only way 
to do this is to select a couple of stations on the earth 
and make simultaneous observations of the same meteor. 
Figure 94 shows how this is done. 

319 




Fig. 94. Height of Meteors. 



ASTRONOMY 

If observers on the earth's surface, at Oi and 2 , see the 
meteor M projected on the celestial sphere near the stars Si 
and S 2) it is clear that we can calculate the height MH 
of the meteor from the known distance Oi0 2 between tjie 
observers, and the angles SiOi0 2 and S 2 2 0i, which are, of 
course, known, if we know the positions of the stars Si and 
S 2 on the sky. 

But there is great difficulty in securing these observations 
in such a manner as to be certain that they apply to the 
same meteor. It is necessary to make the attempt on 
some night when numerous meteors are to be expected; 
and it is then by no means easy to be certain that the ob- 
servations of M from the two stations are really simultaneous, 
and apply to but a single meteoric object. Such as they are, 
observations of this kind indicate an extreme height of 
75 miles for our atmosphere. 

Having thus explained the principal facts about the 
meteors, we come next to the Aerolites. These are stones, 
or pieces of iron mixed with other materials, which fall upon 
the surface of the earth from time to time. There is little 
doubt of their being meteors that actually strike the earth, 
probably on account of unusually large size. For a very 
big meteor would not be entirely consumed, as it were, in its 
passage through the air, and might be attracted down to 
the surface of the earth by the gravitational pull of the planet. 
Those that are completely consumed perhaps fall on the 
earth finally as dust : certain it is that dust, probably 
meteoric, has been found on the surface of ancient Arctic 
ice. This is the so-called " cosmic dust." Many specimens 
of aerolites are preserved in museums. A number have 
actually been seen to fall, so there is no doubt whatever as 
to their origin being outside the earth itself. When seen 

320 



METEORS AND AEROLITES 

at night they exhibit a bright round head, with a luminous 
trail. Occasionally there is an audible explosion. 

The outer surfaces of the aerolitic specimens in our museums 
seem to have been melted, showing the effect of the high 
temperatures produced through their impeded motion in 
the air. Chemically, they contain only elements or sub- 
stances known on the earth. 



321 



CHAPTER XX 

STARSHINE 

In the preceding chapters we have completed a somewhat 
detailed description of the solar system, and are now ready- 
to proceed outward into space, to study the distant uni- 
verse of stars. Astronomers believe that an analogy exists, 
more or less close, between our sun and the stars (p. 6). 
Together with its system of planets, the sun may be regarded 
as a small isolated group suspended in space, and separated 
from other similar groups by distances almost incomparably 
greater than any existing within the solar system itself. 
We may be quite sure that the stars are all excessively dis- 
tant : we are troubled by no doubts in this respect, as were 
the ancients. For we now have the law of gravitation; 
from it we know that if there were a celestial body of any 
kind in space, as massive as the. sun, and not more than ten 
thousand times as far away as the distance separating the 
earth from the sun, — that body would surely reveal its 
existence through observable perturbations (p. 206) pro- 
duced in the motions within our solar system. And this it 
would do, even if it were a dead sun, no longer luminous, 
and quite invisible. To produce gravitational attraction, 
and consequent perturbative effects, merely the presence of 
matter would be necessary, not visible matter. 

Furthermore, actual measurements, to be described 
later, have shown that the nearest fixed star so far observed 
is more than 200,000 times as far from the sun as is the earth. 

322 




Plate 24. The Constellation Serpentarius. 

(From Hevelius' Prodromus Astronomiae, Gedani, 1690.) 



STAItSHINE 

But this last argument is not conclusive, because we can 
measure only visible stars, and the nearest one might con- 
ceivably be non-luminous. This objection, of course, 
does not apply to the gravitational argument. 

We have seen (p. 6) that the stars are classified according 
to their magnitudes, and that this term " magnitude" 
does not here have its usual meaning. It has no relation to 
size or bigness, but simply indicates the degree of luminosity 
or brightness of a star. We shall now consider this matter 
somewhat more in detail. Old Hipparchus (pp. 127, 189, 
297) was the first to divide the stars into magnitude classes ; 
he simply selected arbitrarily the twenty brightest stars 
he could see, and called them first-magnitude stars. He 
then designated as sixth-magnitude all objects that belonged 
at his absolute lower limit of vision, — that he could just 
see, though with difficulty. Stars of intermediate luminosity 
he placed in intermediate classes, also somewhat arbitrarily. 
This gave a rather rough classification ; but it is still in use 
(with some improvements) down to the present day. 

Adopting Hipparchus' magnitude scale, we find the num- 
ber of stars of the various magnitudes situated between the 
north pole of the heavens and the circle of declination 35° 
south of the celestial equator to be as follows : 

1st mag. 14, 3d mag. 152, 5th mag. 854, 
2d mag. 48, 4th mag. 313, 6th mag. 2010, 
Total, 3391. 

The above rough system of classification was replaced by 
a more exact one about 1850. Sir John Herschel had 
remarked from his photometric observations that first- 
magnitude stars average just about 100 times the luminosity 
of sixth-magnitude stars. So it was suggested that we take 
the exact " fifth root " of 100 as the ratio between the lumi- 

323 



ASTRONOMY 

nosities of any two successive star-magnitudes. And this 
ratio is called the " light-ratio." 

In this way, we make the fifth-magnitude stars VlOO 
times as bright as the sixth-magnitude stars; the fourth 
VlOO times as bright as the fifth; etc. Consequently, the 
first-magnitudes would be VlOO X VlOO X VlOO X VlOO 
X VlOO, or 100, times as bright as the sixth-magnitudes, 
in exact accord with Herschel's observation. 

Now the fifth root of 100 is about 2}, so that stars of any 
magnitude are approximately two-and-one-half times as 
bright as those of the next fainter magnitude. To fix a 
definite zero for this scale, it has been decided to select 
certain stars as standards. Thus, Aldebaran is a standard 
first-magnitude : other stars can be compared with it ; their 
light- ratio measured by observation; and the magnitude 
difference then ascertained. 1 This process, of course, some- 
times assigns zero-magnitude, or even a negative magnitude, 
to an exceptionally bright star, like Sirius. On the above 
scale, Sirius actually comes out from photometric observa- 
tions as minus 1.4. The sun's stellar magnitude is about 
— 26, the enormous luminosity being in this case due to 
proximity, not to intrinsic light-giving power. 

Observations of the relative luminosities of stars are made 
with an instrument called an astronomic photometer. The 
ordinary telescope may be so used in a very simple way. 
To estimate a star's brightness, we have only to place dia- 
phragms pierced with holes of various sizes outside the 
object-glass (p. 272) until we find one that will just allow 
us to glimpse the star. It is obvious that this method is 
possible, for if we use successively a series of diaphragms 
with apertures diminishing gradually, we shall in effect be 

1 Note 38, Appendix. 
324 



STARSHINE 

making the telescope smaller and smaller, and there must 
come a time when it has been made so small that it will 
just fail to show the star under observation. From the size 
of the aperture in this last diaphragm, it is possible to cal- 
culate the luminosity of the star. 1 

There are also other, and perhaps better, forms of pho- 
tometers, in which the star under examination is compared 
with an artificial star produced by a light in the observatory 
placed behind a screen having a very small hole. Varying 
the artificial star until it appears of the same luminosity 
as the real one enables the observer to measure accurately 
the brightness of the latter. Magnitudes may also be 
measured photographically. The little dots produced on 
a sensitive plate by prolonged exposure in the telescope 
vary in a sort of proportion to star-magnitudes : the bright 
stars produce larger dots. Therefore a microscopic measure- 
ment of the dot diameters on an astronomic negative enables 
us to estimate star-magnitudes. 

But all astronomic photometric measures are subject to 
considerable error on account of " light-absorption " in 
the terrestrial atmosphere. Some of the stellar light is lost 
in passing through our air. This effect is, of course, smallest 
for stars near the zenith; for there light passes straight 
through the atmospheric layer, and at right angles to it. 
The path through the air is thus the shortest possible. But 
for stars near the horizon the light enters the atmosphere at a 
rather small angle, and its path is much longer, before it 
reaches the observer's eye. Consequently, stars are bright- 
est when they are near the zenith. 

How much is the total light received from the stars? 
This question has been widely studied, but only the very 

1 Note 39, Appendix. 
325 



ASTRONOMY 

roughest results have been obtained. Possibly the whole 
sidereal heavens give about as much light as 2000 stars 
like Vega. This is approximately 3V of full moonlight; 
and it includes the considerable quantity of starlight coming 
from objects below the sixth magnitude, and therefore 
invisible to the unaided eye. 

The heat received from the stars is almost evanescent ; 
only the very slightest indications of it have been rendered 
perceptible by the most delicate thermometric apparatus 
so far invented. 

It is also possible to obtain a very rough comparison 
between the total light actually emitted by the stars and by 
the sun. It is found, for instance, that Vega emits about 
49 times the light sent out by the sun. 1 Other stars give 
similar results : we see, therefore, that our sun is really 
a rather faint star, but not an infinitesimally small one, in 
comparison with Vega. 

There exists still another very remarkable phenomenon 
in connection with stellar luminosity, — its variability. For 
it must not be supposed that the brightness of all stars is 
strictly constant : many increase or diminish their light from 
time to time; and these are called " variable stars." There 
are several distinct kinds. Some vary their light steadily, 
ever increasing or diminishing it ; others show no regularity 
whatever in their rise and fall; and a few, the " temporary 
stars," or novce, appear suddenly in the heavens, and last 
but a short time. Finally, there are stars that wax and wane 
with more or less accurate periodic regularity ; and wonderful 
variables, in which changes are caused by some form of 
eclipse phenomenon. Of these last the best known is the 
star called Algol (the Demon). 



1 Note 40, Appendix. 
326 






STAESHINE 

The number of stars whose light alters steadily in one 
direction is very small : demonstrated permanent variations 
of brilliancy since the time of Hipparchus are extremely in- 
frequent. But we have records that Eratosthenes saw @ 
Librae brighter than Antares, though the contrary is surely 
the fact at present. Similarly, in 1603, Bayer recorded 
Castor as brighter than Pollux, in the constellation Gemini ; 
but now Pollux gives us more light than Castor. 

The most prominent irregular variable is 77, in the con- 
stellation Argo Navis, in the southern celestial hemisphere. 
Sir John Herschel saw it as bright as Sirius in 1843, on the 
occasion of a visit to the Cape of Good Hope. It has been 
of the seventh magnitude since 1865, and is still in process 
of change. 

Temporary stars have blazed up about eighteen times 
since men began to write their records of the skies. The 
most famous is Tycho Brahe's star of 1572, which was 
brighter than any other star, and lasted only sixteen months 
all together. This is the star that first interested Tycho in 
astronomy : it is reported that he refused for a long time to 
describe his observations, because he thought it beneath the 
dignity of a Danish nobleman to write a book. 

A peculiarly interesting recent object was nova Auriga 
(the new star in the constellation Auriga), which appeared in 
1891. It has been variously explained as the result of 
heat engendered by a collision between two dark stars, or as 
an explosion occurring in a single dark body. 

Another important new star was found in the constellation 
Perseus in 1901. Within a few days after discovery its 
brightness grew to the first magnitude, but it faded again 
with almost equal celerity. It developed a surrounding 
nebulosity after a time ; and is all together one of the most 

327 



ASTRONOMY 

wonderful astronomic objects ever observed. The accom- 
panying Plate 25 shows the manner in which this nebulosity- 
grew in size. If this growth is the result of motion outward 
from the central star, the velocity must have been incredibly 
rapid. What cosmic process or catastrophe there occurred 
before our eyes, we can neither describe fully nor attempt to 
explain. 

The periodic variable stars are of three kinds. First, 
those like the star Mira (the Wonder) in the constellation 
Cetus, which is ordinarily invisible without a telescope, 
but increases rather suddenly every eleven months to the 
third magnitude, when it is of course a naked-eye star. The 
second variety of these variables includes stars like /3, in the 
constellation Lyra, with changes completed in a very short 
period; and with the alteration of light apparently com- 
pounded of two different variations superposed. And the 
third class of these stars are like Algol, in the constellation 
Perseus, which, at regular intervals, undergoes a partial 
eclipse. 

A possible explanation of regular variations might be of- 
fered from the analogy of sunspots. Certain stars may have 
very large permanent spot zones that are carried around by 
axial rotation ; and when these are turned toward the earth 
we may properly expect for a time a considerable diminution 
of light. 

The eclipse theory for the Algol stars is quite old ; but it 
was not proved correct in a convincing way until 1889, when 
Vogel successfully observed the spectrum according to the 
Doppler principle (p. 284). He showed that the visible 
star Algol has a velocity of recession from the earth before 
its period of minimum luminosity, and an equal velocity of 
approach after such minimum. The explanation now pos- 

328 



STARSHINE 



tulates (Fig. 95) a dark but massive companion to the visible 
star, and both revolving in nearly circular orbits about 
their common center of gravity. The orbit plane is sup- 
posed to be directed towards the solar system, so that we see 
the orbits edgewise. The dark body is supposed to be 
smaller in size than the luminous or visible one. 

In the course of their orbital revolutions, there will come a 
time when the smaller dark body will be at D, and the lumi- 
nous body at L. An observer on the earth, in the direction 
shown by the straight arrow, 
will see the luminous body _j.»<£i>2**a 

partly covered or eclipsed 
by the smaller dark body. 
As soon as the eclipse is 
over, the luminous body 
will regain its full brilliancy, 
as in the positions D' and U. 

From a combination of 
the velocities of approach 
and recession, observed 
spectroscopically, with the 
known light-variations of 

Algol, Vogel was able to prove that the distance between the 
two bodies is 3J million miles, and their diameters respec- 
tively 0.8 and 1.1 million miles. It will thus be seen that 
they are so near each other as to be almost in rolling con- 
tact ; but there exists no cosmic law preventing the occur- 
rence of orbital motion of this kind. 

The combined mass of the two bodies Vogel found to be 
about | that of the sun ; his calculations were made by a 
process analogous to the method of determining a planet's 
mass from the observed orbital period of its satellite (p. 204). 

329 




Fig. 95. Explanation of Algol Stars. 



ASTRONOMY 

The more complicated light-variations of certain other 
stars may also be explained on the eclipse theory, the orbital 
planes being supposed inclined to the direction of the earth, 
and neither body dark. If both are luminous, there will 
still be a diminution of light during the eclipses. For one 
luminous body will then appear superposed on the other, 
and the total light will be the same as the larger body would 
emit alone. 

The next thing we have to consider is the important 
question of stellar distances from the solar system. After all, 
this is in a way the most interesting matter we have to dis- 
cuss in connection with the stars, for the question at issue 

really relates to the 
actual dimensions 
of the magnificent 
sidereal universe. 

In the first place, 
let us once more 
define " stellar par- 
allax " (p. 192). 

Fig. 96. Stellar Parallax. 

Solar parallax (p. 
260) has been explained to be half the earth's angular diam- 
eter, as seen from the sun. In a similar way, stellar par- 
allax is the angle SS'E, in Fig. 96, between two lines, one 
drawn from the star S' to the sun S, and the other from the 
star S', tangent to the earth's annual orbit around the sun 
at E. In other words, when the earth reaches such a posi- 
tion in its orbit around the sun that there is a right angle at 
the earth E, between the sun S and the star S' } then the angle 
SS'E is the star's parallax. 

We are compelled to use the radius of the earth's orbit 
in defining stellar parallax, whereas we used the radius of the 

330 




STARSHINE 

earth itself in the case of solar parallax. For the earth's own 
radius is far too small to subtend an angle at all appreciable, 
if the vertex is situated at the vast distance of the stars. 
Even when we use the orbital radius, the very nearest star, 
so far as we now know, has a parallax angle of only three- 
quarters of a second of arc. 

It is clear, from Fig. 96, that the star's parallax angle 
really equals the difference in direction of the star as seen 
from the earth, and as it would be seen by a supposed ob- 
server on the sun. As the earth goes around its orbit, this 
parallactic displacement, or change of the star's direction 
due to the observer's being on the earth instead of the sun — 
this parallactic displacement must change its direction. 
At intervals of six months, during which the earth traverses 
one-half of its orbit, the displacement reaches equal amounts, 
but opposite directions. In the interval, there is a constant 
change in the direction of the displacement ; so that, if a star 
is projected on the sky at a point perpendicular to the plane 
of the earth's orbit, it will appear to describe in a year a 
little circle on the sky, which is a miniature replica of the 
earth's own orbit around the sun. 1 A star in the plane of 
the earth's orbit will simply appear to swing back and forth 
through the year in a short, straight line, instead of describing 
a circle. Stars in intermediate positions will have apparent 
" parallactic orbits," which will be small ovals, intermediate 
in form between the circle and the straight line. 

The measurement of a star's parallax is therefore nothing 
more than a measurement of the form and angular diameter 
of its little apparent parallactic orbit on the sky. This may 
be measured by an " absolute" method, or a " differential " 

1 A star so situated is, of course, at the "pole" of the ecliptic circle on 
the sky. 

331 



ASTRONOMY 

one. The absolute method requires a determination of the 
star's right-ascension and declination at various dates 
during a year. These being located on a celestial chart 
give the parallactic orbit at once. But the method is 
practically of little value, because we possess no instruments 
capable of measuring these declinations, etc., within the 
small fraction of a second of arc which is here necessary. 

The differential method is better, since it enables us to use 
a micrometer (p. 276) as illustrated in Fig. 97. The parallactic 
orbit of a star is here shown as an ellipse or oval (p. 331). Si 
and S 2 are apparent positions of the star in its parallactic 
orbit on two different dates. Si and S 2 are 
two small stars in the vicinity. The observer 
measures, on various dates through the year, 
the small angular distances, from the parallax 
star to the small stars Si and S 2 . These 
Fig. 97. Differen- latter, on account of their minuteness, may 
reasonably be supposed situated at practically 
an infinite distance from us, and therefore to have no appre- 
ciable parallaxes of their own, and no parallactic orbits. So 
the measures determine a series of points on the parallactic 
orbit ; and, from the size of the orbit, the parallax of the 
star S. 

It is clear that this method should bring out a value 
of the parallax possessing a high degree of precision : 
and microscopic measures of an astronomic photograph 
may here replace actual visual micrometric work at 
the telescope with advantage. If the small stars are not 
really at an infinite distance, the differential method fur- 
nishes what may be called " relative parallaxes"; or, in 
a sense, the parallax excess of the star under examination 
over the small stars supposed infinitely distant. The first 

332 




STARSHINE 

successful stellar parallax measurement was made by Bessel 
in 1838. He used the differential method, and applied 
it to a star of moderate magnitude in the constellation 
Cygnus (cf. p, 192). 

When we come to translate stellar parallax measures into 
terms of linear distances, we arrive at numbers so large as 
to be unmanageable. For this reason astronomers have 
invented a new and large linear unit to be used in sidereal 
measurements. It is called the " light-year," and is defined 
as the linear distance light would travel in a year. As the 
velocity of light is about 183,000 miles per second, the light- 
year, in miles, amounts to 183,000 multiplied by the number 
of seconds in a year. It is therefore an enormous unit of 
distance ; but it is none too large for use in describing stellar 
distances in space. Its length is about 60,000 times the dis- 
tance from the earth to the sun, and corresponds to a paral- 
lax of 3 J seconds of arc. 

Closely connected with the above subject of stellar dis- 
tances is the question of the stars' motions. We have 
already seen (p. 7) that the objects called fixed stars are 
not really fixed in space. They are all actually drifting 
across the sky; it is only because of their vast distance 
that their motions seem to us small and slow. In reality, 
their velocities are of the same order of magnitude as those 
existing among the planets of our solar system : but to the 
rough instruments of the ancients these motions remained 
unrevealed ; and so the stars received their designation of 
" fixed," to distinguish them from the wandering planets. 
And within the last half-century it has become possible 
to do even more than merely measure this stellar drift 
across the face of the sky ; the drift which shows itself as a 
change of the star's right-ascensions and declinations. As we 

333 



ASTRONOMY 

have already seen (p. 284), we can now also evaluate, with 
the spectroscope, stellar velocities of motion in the "line of 
sight"; or, in other words, the star's linear velocities in a 
direction perpendicular to the celestial sphere. 

Now changes in a star's right-ascension and declination do 
not necessarily prove the existence of motion. For the pre- 
cession of the equinoxes (p. 126) moves the point from which 
we count right-ascensions, and it also shifts the celestial 
equator from which we count declinations. Since right- 
ascension is angular distance from the vernal equinox, 
measured on the equator, and declination angular dis- 
tance from the equator, it is clear that precessional changes 
of equinox and equator will change both these quantities. 

But all precessional effects can be calculated easily: 
and even after these effects have been eliminated from our 
observations by a suitable process of calculation, we still 
find small " residual" changes of right-ascension and decli- 
nation. These may be ascribed wholly or in part to real 
motions of the stars. It is this residual motion that is called 
a star's " proper motion." This term is now a century old 
in astronomy : it is applied only to motion across the celestial 
sphere; not to motion in the line of sight, revealed spec- 
troscopically. The latter has been separately denominated 
" radial velocity." Proper motion is measured in seconds of 
arc per annum ; radial velocity in linear miles per second. 

Less than two hundred stars have proper motions as large 
as one second of arc per annum. The largest known mo- 
tion of the kind belongs to a little star numbered V 243 in a 
great catalogue of stars made at Cordova in South America. 
It drifts nearly nine seconds annually. The next largest and, 
until 1898, the largest known belongs to a star numbered 
1830 in a catalogue observed by Groombridge in England 

334 



STARSHINE 



about the year 1790. This star is often called the " run- 
away" ; it travels seven seconds in a year. 

The relation of proper motion and radial velocity is indi- 
cated in Fig. 98. If a star at Si moves to S 2 in a unit of time 
(a year, let us say), and if the solar system is at E, we can 
draw S1S1 perpendicular to SiE, to indicate the star's appar- 
ent motion across the celestial sphere as seen from E. In the 
same unit of time the star will have receded from the earth by 
a distance &&'; and this will 
represent its annual radial ve- 
locity. The true motion of the 
star, SiS 2 , may be regarded as 
really made up of two parts, 
— the radial motion and the 
proper motion. Now that we 
are able to measure these 
radial velocities, it is at last 
possible to ascertain from ob- 
servation both parts of the dis- 
tance S1S2, which is the star's 
true motion in a unit of time. 
Before we had the spectro- 
scope and Doppler's principle, we never knew, or could know 
more than the one part S1S1. These considerations empha- 
size the importance of the spectroscope in sidereal research : 
it has not only created a celestial chemistry ; it has also given 
us new and essential knowledge in the oldest department of 
astronomic science, — the theory of sidereal motions. 

Not infrequently it happens that a certain number of 
stars are found to have proper motions, and probably real 
motions in space, nearly identical, both in amount and direc- 
tion. For instance, Proctor pointed out some years ago 

335 




Fig. 98. 



Proper Motion and Radial 
Velocity. 



ASTRONOMY 

that five of the bright stars in the Dipper (p. 52) probably 
possess such a community of motion. There can be little 
doubt that they are proceeding through space in parallel 
lines, and that they belong together. The same is true of a 
number of the brighter stars in the Pleiades group. They 
are not merely an apparent group, but a real cluster. 

A very surprising thing, discovered in 1909, is that Sirius 
is also probably a member of the Dipper group of stars. If 
this be so, the group in question is not a distant congeries 
passing us at a distance, perhaps, of billions of miles; but 
it is passing so close that we are actually now within the 
drifting group of stars itself. This follows from the fact 
that Sirius is in quite a different part of the sky from the 
Dipper region. 

We have seen quite enough to make clear the high value 
to science of these very modern measures of radial velocity : 
unfortunately, it has not been found possible as yet to apply 
the method to faint stars, because their light is not sufficient 
to give a measurable spectral image. This class of work was 
first attempted by Huggins in 1867 ; and he began by measur- 
ing the radial velocity of Sirius, the brightest of the stars, 
with a visual telescope and spectroscope. It was not until 
Vogel began to use photography in 1888 that any consider- 
able extension of the process became possible. Fainter 
stars then first became observable, for the exposure of a 
photographic plate can be lengthened within reasonable 
limits, so as to give even a small quantity of light time to 
impress a spectral image on the plate. 

Good examples of radial velocities are found in the star 
a Carinse, receding from the solar system 17 miles per 
second ; and o Ceti, receding 54 miles per second. As the 
earth's own orbital velocity around the sun is about 18| 

336 






I 

1 

■ _ 

II 






o {ft 



s ^ 




STARSHINE 

miles per second, we see that the stellar motions are only of 
the same order of velocity as those existing within the con- 
fines of the solar system. 

Spectrum photographs of the above stars are shown in the 
lower part of Plate 26. In each case the stellar spectrum 
is placed between two artificial spectra, produced in the 
observatory for comparison. It will be seen that the stellar 
spectral lines of both stars are displaced in the same direction 
with respect to the artificial spectra, because both stars 
are receding; but the displacement is much greater in the 
case of o Ceti, on account of its greater velocity of recession. 
The upper photograph of Plate 26 is a comparison of the lunar 
spectrum with that of Saturn and its ring (cf. p. 245). It 
shows, as it should, that the outer part of the ring is moving 
slower than the inner part. 

This matter of stellar spectroscopy was first taken up by 
Huggins to study the chemical composition of the stars, 
though it led him also to the measurement of radial veloci- 
ties, as we have seen. At about the same time, Secchi 
undertook similar investigations, and to him we owe a sort 
of classification of stellar spectra, as follows : 

1. White and blue stars, with strong evidence of hydrogen. 
Examples are Sirius and Vega. These stars are believed to 
be in an early stage of cosmic development. 

2. Solar stars, showing in their spectra many dark lines 
due to absorption, as in the solar spectrum. Capella is 
one of those stars; they are supposed to be somewhat 
older than those showing the hydrogen lines. 

3. Red stars, with spectra showing dark, broadened lines. 

4. Faint red stars, probably very old ; the spectra having 
a few bright lines. 

All the spectroscopic observations indicate a stellar 
z 337 



ASTRONOMY 

chemistry similar to that of the solar system. The entire 
sidereal universe seems to contain but one set of chemical 
elements; and these are very widely distributed. So we 
see once more that our sun is a star; and if the other 
stars are traveling through space, our sun should also be in 
motion, carrying its planets with it, much as the earth 
moves in its orbit around the sun and carries the moon with 
it. Analogy would lead us to expect such a solar motion. 

Of course, the simplest way to study this question is to 
examine the radial velocities of a large number of stars. Sup- 
pose we find that those near a certain point on the sky are 
approaching us; those near the opposite point receding; 
and those halfway between, neither approaching nor reced- 
ing. Then we may conclude that this is merely a result of 
the solar system's own motion ; and that we are approaching 
the stars projected near the point of the sky where they 
seem to be approaching us. Towards this point on the sky, 
then, the solar motion is for the moment directed. 1 

Campbell made such a research a few years ago, using 
spectroscopic results derived from 280 stars. The point 
on the sky indicated by his work is not very far from the 
first-magnitude star Vega. It is called the "apex of the 
sun's way." Naturally, Campbell used only bright stars 
whose spectra could be observed ; and of course brightness 
indicates nearness, other things being equal. It may there- 
fore be a fact that stars having a common drift with the sun 
predominate in Campbell's series of observations ; and if so, 
this might partly invalidate his result. But however this 
may be, he finds the region near Vega to contain the apex, 
and 13 miles per second as the " cosmic linear velocity" of 
the solar system. 

% > Note 41, Appendix. 

338 



STARSHINE 

Another fact that may cause a slight error in such an in- 
vestigation as the foregoing is the necessity of making 
some assumption as to the average real motions of the stars 
whose radial velocities are observed. For we do not find all 
stars near the apex approaching us; only a preponderance 
of motions of approach. So astronomers assume that in 
the average of so large a number of stars (in this case, 280) 
there will be as many motions in any one direction as in any 
other. Therefore, a preponderance of motions of approach 
near the apex must be due to solar motion, not to motion of 
the stars themselves. 

It is singular that as far back as 1783 Sir William Herschel 
obtained almost the same result from a discussion of the 
proper motions (p. 334) of various stars across the sky. His 
method is well illus- 
trated in Fig. 99. If 
there are two lamp- *v 
posts, Li and L 2 , on 
opposite sides of a ^ U2 

., , ,. Fig. 99. Determination of the Apex (Herschel). 

street, the angular dis- 
tance between them will seem much larger when they are 
viewed from H 2 than from Hi. A person walking from Hi 
to H 2 will see this increase of the angular distance. Apply- 
ing this principle to the sky, Herschel concluded that near 
the apex the constellations must be opening out, as we ap- 
proach, and at the opposite point of the sky they must 
be closing in. 

In other words, near the apex stellar proper motions directed 
away from that point must predominate ; and near the op- 
posite point, called the " anti-apex," proper motions directed 
towards the critical spot must be most in evidence. Herschel 
had at his disposal the measured proper motions of only 

339 




ASTRONOMY 

thirteen stars all together. Yet with the insight of rare 
genius, he so sifted this meager evidence that he was able 
to find the right-ascension and declination of the apex 
with some approximation to correctness. 

Later investigators have of course repeated this work with 
much more elaborate modern material at command. They 
find a result in very fair accord with the spectroscopic one. 
But they also find this important peculiarity. If the proper 
motions are divided into groups, and the calculations made 
separately with stars of large and small proper motions, 
the apex comes out farther south on the sky for the stars of 
large proper motions. 

Now it is evident that any investigation of this kind must 
assume that if there were no solar motion, the stellar proper 
motions would be quite casual, and free from any tendency 
to congregate in direction towards or from any apical point. 
But such a tendency in the stars themselves is indicated 
by the peculiar result just mentioned. It is clear that the 
large proper motion stars must have a common path of 
their own. But largeness of proper motion should indicate 
nearness to us, other things being equal ; for at a sufficient 
distance, even large proper motions would shrink into ap- 
parent nothingness. Therefore it is within the bounds of 
possibility that our sun is a member of a drifting stream of 
stars, to which, in general, the large proper motion stars 
belong also. 

In the light of the above discussion of cosmic motions of the 
sun and stars, as well as stellar distances, it is possible to con- 
sider an interesting special problem which may be solved 
approximately with modern observational data. We have 
seen that the solar system is moving toward Vega at the 
rate of 13 miles per second (p. 338). Observations of Vega's 

340 



STARSHINE 

own radial velocity indicate that it is itself receding from us 
at the rate of 3 miles per second ; so we are overtaking it at 
the rate of 10 miles per second. But the number of seconds 
in a year is, approximately, 365 X 24 X 60 X 60 ; and the 
actual annual approach of the two stars, therefore, 365 X 24 
X 60 X 60 X 10 miles. The parallax of Vega has been 
measured; it is O/'ll. From the parallax, the distance 
between the solar system and Vega may be computed, 1 and 
it comes out, approximately : 

93000000 X 200000 » 

o.n miles - 

To ascertain the time in years required by the solar 
system to reach the position occupied by Vega, we must 
divide the distance of Vega by the rate of approach per year. 
We thus obtain, for the number of years required to over- 
take Vega in space : 

93000000 X 200000 1 

X 



0.11 " 365 X 24 X 60 X 60 X 10 

or, approximately, 560,000 years ; and after the lapse of that 
period of time, the solar system should reach Vega. 

But in that interval Vega will have moved, too, for it 
has a proper motion across the sky of 0."5 per year, which is 
about four times its parallax angle. Figure 100 will explain 
this state of affairs. At a certain moment, the sun and 
earth are at S and E, with Vega at V\. Then, by definition, 
the angle EViS is Vega's parallax. At the end of a year, 
the earth will be back at E, very nearly, but Vega will be 
seen at V 2 , because its proper motion will have carried it 
across the sky, as seen from the solar system, through the 
angle ViSV*. Since this angle is four times the parallax 

1 Note 42, Appendix. 
341 



ASTRONOMY 

angle EViS, it must follow, approximately, that ViV 2 is 
four times ES. Therefore, ViV 2 is 4 X 93,000,000 miles, or 
372,000,000 miles. 

So we see that when the sun reaches the point where 
Vega should be in 560,000 years, Vega will have moved at 




Fig. 100. Vega's Parallax and Proper Motion. 

least 372,000,000 X 560,000 miles, and this is about as 

near as we shall ever approach Vega. What will be Vega's 

parallax at that time? We can answer this question by 

comparing the present distance of Vega, which we have 

found to be : 

93000000 X 200000 ., 
Qjj nnles, 

with its distance in 560,000 years as just obtained. The 
two numbers are not far from equal, and therefore Vega's 
future parallax will not be far from its present parallax. 
So there is no danger of a cosmic collision with Vega, so far 
as we may judge from the above rough calculation. 

Our present discussion would not be complete without a 
brief account of certain quite recent researches, made 
principally by Kapteyn. His idea is that we need extensive 
statistical knowledge of stellar distribution, more than direct 
measures of a few parallaxes. He therefore undertakes to 
compute the average parallax of the stars of any given magni- 

342 



STARSHINE 

tude, separating the stars thus by magnitudes for the obvious 
reason that the fainter ones may be expected to average 
greater distances than the brighter ones, and therefore 
smaller parallaxes. Figure 101 shows his method of attack 
upon this problem. The arrow SS' is intended to represent 
the annual proper motion of a star S across the face of the 
sky, the arrow, of course, indicating both the direction and 
quantity of such motion. The position of the apex on the 
sky is shown at A. The line 

S'Si is drawn perpendicular / 

to SA, and the smaller arrow / 

SSi then shows how much / 

the proper motion SS' carried / 

the star away from the apex. c , / 

In fact, the proper motion 
SS' may be regarded as com- 
pounded of two motions : 

Fig. 101. Kapteyn's Researches. 

ooi , which affects the an- 
gular distance from the star to the apex; and Si'S', which 
does not affect that distance, being at right angles to SA. 

Now Kapteyn " resolves" (as it is called) all known 
proper motions into two such " components," one directed 
away from the apex, the other at right angles to the first. 
But we have already seen (p. 339) that the effect of the solar 
system's own motion in space is to open out the constella- 
tions near the apex ; therefore SSi, the star's proper motion 
component away from the apex, must include the effect 
of the solar system's motion ; but Si'S', the other component, 
is free from such effect. In the general average of a large 
number of stars of any given magnitude, the two components 
should be equal, if the sun were at rest. For the effect of 
the solar motion would then disappear ; and we may assume 

343 




ASTRONOMY 

that the star's own motions are as likely to be in one direc- 
tion as another, so that the average components would 
balance, as it were. It follows that any difference of the 
two components, derived from observed averages, must be 
an effect of the sun's motion alone. 

Now this average difference is expressed in seconds of arc, 
being an observed angular proper motion across the sky. 
And we know the linear velocity of the solar system toward 
the apex to be 13 miles per second (p. 338), or 13 X 365 X 24 
X 60 X 60 miles annually. Figure 102 shows the further 




Fig. 102. Kapteyn's Researches. 

procedure. The arrow SiS 2 represents the solar system's 
motion toward the apex, in a year. Kapteyn's " average 
star" is shown at $ 3 . The little angle SiS s S 2 is the average 
star's proper motion component away from the apex, or 
the above-mentioned observed difference of the two com- 
ponents. Knowing, now, this little angle, and the linear 
velocity SiS 2) we can calculate SiS 3 , or the average star's 
distance from the solar system Si. 

This beautiful method of investigation has enabled 
Kapteyn to obtain a table of approximate stellar distances. 
He gives the following results, for two types of stars sepa- 
rately (cf. p. 337) : I, bluish white stars, like Sirius ; II, solar 
stars, like Capella. The distances are expressed in light- 
years (p. 333), three of which correspond approximately to 
a parallax of one second of arc. The table shows that the 

344 



STARSHINE 

solar stars are decidedly nearer to us than are the Sirian 
stars. 

Table of Kapteyn's Distances 

(In light-years) 







Type 




Star's Magnitude 








I 


II 


1 


101 






43 


2 


130 






56 


3 


166 






71 


4 


213 






91 


5 


273 






117 


10 


948 






405 


15 


3270 






1404 



Kapteyn has also obtained some further interesting results 
as to stellar distribution. He uses a "unit of sidereal space," 
and for this space unit he imagines a cosmic sphere of "unit 
radius/' which he defines as a sphere such that a star on 
its surface would have a parallax of one second of arc to 
an observer at its center. The length of this radius would 
be about three light-years. Then, since parallax angles are 
inversely proportional to distances, it follows that stars 
whose parallaxes are greater than 0."20 are all within a 
sphere whose radius is 5 units. About 20 stars with such 
parallaxes are known ; and we may assume that these are 
probably all that exist. 

Now the volumes of spheres are proportional to the 
cubes of their radii ; so that the above-mentioned sphere with 
radius 5 must have a volume of 125. And as there are 20 
stars in it, there must exist about one star to each six units 
of space; and this is the approximate "star-density" in the 
cosmic vicinity of the solar system. This conclusion may 

345 



ASTRONOMY 

not be very accurate ; but it constitutes a most important 
addition to our knowledge of sidereal astronomy, since even 
a rough approximation is better than a total absence of 
information. 

And Kapteyn has been able to proceed a step farther. 
Having found the average distance of stars of any given 
magnitude, and knowing the average proper motions and 
radial velocities as well, he has been able to compute the 
actual average linear velocities of the stars in space; and 
he finds them to be somewhat less than twice the cosmic 
velocity of the solar system. 1 Therefore the stars' average 
speed is about 26 miles per second; or, in a year, about 
seven times the distance between the earth and sun. But 
the sidereal unit, or radius of the unit sphere, is about 
200,000 times the distance from earth to sun ; so the stars 
move a sidereal unit, on the average, in 27,000 years. 

Now we have found that, on the average, about one star 
exists in each six units of space. From this it may be com- 
puted, according to the theory of probabilities, that the 
average distance between the stars is about 3.5 linear 
sidereal units. Therefore the stars move through a distance 
equal to the 3.5 units that separate them in about 3.5 X 
27,000, or 100,000 years, on the average. It follows from 
all these considerations that stars will approach each other 
infrequently, even within astronomical proximities, enormous 
as these are. 

The above conclusions all relate to averages ; and we know 

1 Knowing the star's parallax, or distance, and the angular annual 
proper motion across the sky, we compute the linear velocity component 
parallel to the celestial sphere in the same way that we obtain a planet's 
linear diameter from its measured angular diameter. Then, knowing the 
radial velocity, and the linear velocity at right angles to it, as computed 
from the proper motion, we can finally calculate the actual velocity. 

346 



STARSHINE 

one or two stars that probably have far greater velocities 
than the average. For instance, the star 1830 Groombridge 
(p. 334) has a velocity of perhaps 140 miles per second. We 
must conclude that this particular star is passing through 
our sidereal universe, and will leave it altogether in a few 
million years. But in general, from what has been said, 
it would appear that the stars in our universe are much like 
the molecules of a gas, as indicated in the kinetic theory of 
gases. The difference between a glass vessel full of gas and a 
universe full of stars is merely one of scale. In either case, 
each star or molecule moves in a more or less straight line 
of random direction, until or unless a couple of them happen 
to collide. In both cases, such collisions are extremely fre- 
quent : only, in the gas, the word " frequent " signifies a very 
minute fraction of a second ; in stellar space, the same word 
may mean centuries. 

Perhaps this kinetic theory of stars would undergo some 
material modification if we admit that all observed stellar 
motions are not necessarily random ones, and that there may 
be star-groups of common motion, and star-streams of vast 
extent. But however this may be, these magnificent 
researches are all inspiring in a high degree : it is extraordi- 
nary that such can still be made in our own day in the oldest 
and most completely perfected domain of human knowledge, 
the science of astronomy. 

Not infrequent in the sky are the " binary" stars; 1 twin 
suns, they have been called (cf. p. 9). Each component star 
of such a pair moves around the center of gravity of both in an 
oval orbit, just as the earth and moon (p. 174) move around 
their center of gravity. The orbits are studied with a special 
micrometer (p. 276), with which astronomers can measure 

1 The lower part of Plate 27 is a photograph of a binary. 
347 



ASTRONOMY 

the angular distance between the two components in seconds 
of arc, and also the angle between the line joining them on 
the sky and a line drawn from the principal one to the celes- 
tial pole. Thus, in Fig. 103, SiP is an arc of a great circle 
i? imagined on the sky, joining the principal 

component star Si and the pole. We then 
measure the small angular distance SiS 2 , be- 
tween the two components of the double star, 
and also the angle PSiS 2 . The latter is called 
the " position angle." If we continue these 
Fig. 103. Binary measures, at intervals, for a number of years, 
star- and then draw the resulting orbital curve, we 

find it to be an oval or ellipse, often very much flattened. 
But this is only an apparent orbit ; for what we see is the 
real orbit, projected on the celestial sphere. It is only when 
the plane of the binary star's orbit happens to be perpen- 
dicular to our sight-line that the apparent orbit coincides with 
the real one. But it is always possible to calculate the 
location of the true orbital plane ; and, in fact, the elements 
of the real orbit, by applying Kepler's laws of motion (p. 187) 
to the apparent orbit. Yet, even after this has been done, we 
have only a " relative" orbit, representing the motion of 
one component star of the binary pair with respect to the 
other. 

For a very few binaries, the actual orbit of both compo- 
nents has been separately determined, by means of micro- 
metric comparisons with neighboring small stars. And 
when we are so fortunate as to know also the parallax of 
the binary, we can calculate the linear dimensions of the 
orbits in miles. Without the parallax, we can know only 
the angular dimensions of these orbits in seconds of arc. 
The relation of the two dimensions is like the relation of the 

348 




Photo by Barnard. Photo Mt. Wilson Observatory. 

Plate 27. A Star Cluster in Hercules and the Double Star Krueger 60. 



STARSHINE 

angular diameter of the sun to its real linear^ diameter 
(p. 118). In the few cases for which such researches have 
been carried out with success, the linear size of the orbits 
appears to be comparable with the orbital radii we find in 
the solar system. So we conclude that the binary systems 
are not extremely large, speaking cosmically. 

Certain binary stars have been recognized as such by 
spectroscopic instead of micrometric observations. We have 
already described VogePs discoveries with regard to Algol 
(p. 328), where the binary character of the star was betrayed 
by an observed periodic change in the direction of its radial 
velocity. But Pickering also found that certain spectra 
photographed with a slitless spectroscope' (p. 285) showed 
a periodical doubling of the liues. Ordinarily single, they 
became double at uniform intervals of time. Pickering 
explained this correctly as indicating a binary system, in 
which, unlike Algol, the components are both luminous stars. 
When one component is approaching us, and the other reced- 
ing, in consequence of their orbital motions, the spectral lines 
are displaced in opposite directions, according to the Doppler 
principle, and we get two separate spectra and two sets of 
lines. When both components are moving at right angles 
to the sight-line, as they will do in another part of their 
orbit, the two spectra are superposed, and we get one set of 
lines only. It is remarkable that we can thus separate a 
pair of stars with certainty, although they appear so near 
each other on the sky that the most powerful telescope shows 
but a single object at all times. This was a great triumph 
for the spectroscope; the observation, together with the 
correct explanation of it, will surely have a place in the 
classic annals of astronomy. 

It is interesting to note that we can calculate also the 

349 



ASTRONOMY 

masses of these binary stars for which both orbits and paral- 
laxes have been observed. As we have just seen, we then 
know the linear dimensions of the orbit in miles, and so we 
can apply the method used (p. 204) for obtaining the mass of 
a planet having an observable satellite. 1 The masses of 
these binaries, in the few cases where they have become 
known, are found to be of the same order of magnitude as the 
sun's mass. 

Before leaving the subject of binary stars, it may be of 
interest to touch on one possible theory as to their origin. 
It is not now believed by astronomers that the Laplacian 
theory of celestial development (p. 235) is the only pos- 
sible or even probable one. For the Laplacian idea leads 
to a single central sun, with many planets of far smaller 
size than the sun. But it is possible that an original 
whirling nebula may have undergone changes more or less 
approximating the formation of two nuclei. These, 
revolving, gave rise, first, to an egg-shaped, — later, a 
dumb-bell shaped, — revolving body. The latter, finally 
separating, should produce twin suns, at first revolving with 
their surfaces almost in contact. Such a condition might 
even explain some of the peculiar light-changes of certain 
variable stars. Later, there might arise perturbative 
action, similar to the tidal effects produced between the moon 
and terrestrial oceans. These would drive the two bodies 
farther apart (cf. p. 258), and possibly lead to a visible binary 
star. Nor is there any objection to our imagining some of 
these distant suns to be attended by planets. Only, if 
such planets are no larger than Jupiter, we could not possibly 
hope to see them with the most powerful of our telescopes. 

It is an easy transition from the binary systems to those 

1 Note 43, Appendix. 
350 




Photo by Barnard. 



Plate 28. The Pleiades. 



STARSHINE 

still more wonderful, in which three or more incandescent 
suns revolve about their common center of gravity in plain 
view of the telescope. For instance, the constellation Lyra 
contains an important double star, the " double double/ ' 
in which each component is itself a binary, forming all to- 
gether a quadruple star. 

And in addition to these "multiple stars," we find also 
various " clusters. " Some contain comparatively few stars, 
spread over quite a considerable bit of the sky. The 
Pleiades group is a famous cluster of this kind. Two views 
of it are shown in the accompanying Plate 28 : one con- 
tains the stars only ; the other, made with a large telescope, 
indicates that most of these stars are still surrounded with 
nebulosities. Here and there we can see a nebulous lane 
running from one star to the next ; nor have these peculiar 
formations ever been thoroughly explained. There are 
also other clusters, like the close-packed globular one in 
the constellation Hercules (Plate 27, p. 349), consisting of 
many thousand stars separated from each other on the sky 
by very small angular distances only. 

We have two sure facts to indicate that the clusters are 
single objects, and not mere fortuitous groupings of stars, 
unconnected with each other, situated at all sorts of distances 
from the solar system, and appearing close together because 
they happen to be projected near a single point of the sky. 
First, in the Pleiades group, it is known that over forty stars 
have practically identical proper motions in the same direc- 
tion on the sky, pointing to a community of real motion in 
space. And secondly, many close clusters have been found 
to contain a most unusually large percentage of variable 
stars, again indicating a community of origin for the whole 
cluster. 

351 



ASTRONOMY 

Unfortunately, it has not yet been possible to measure 
the parallax of any cluster. We can but make guesses at 
their distance from the solar system, rough estimates based 
on the size of their proper motions. These are so small that 
we must assume the clusters to be very distant, — probably not 
less than 400 light-years. If removed to that distance, our 
own sun would give us no more light than a star of the 
eleventh magnitude. It follows that the clustered stars 
are perhaps comparable in size with the sun, for they, too, 
average the eleventh magnitude, more or less. 

With the above estimate of distance, we can also estimate 
the linear size of the clusters from their angular diameter, 
in the usual way ; and we find them to be about two light- 
years in diameter. If such a cluster contained 10,000 stars, 
the average distance from one to another would be about 
25,000 times the distance from the earth to the sun. At 
such distances gravitation would not be strong enough to 
bring all the constituent stars under the influence of a central 
force : it would not even produce velocities of interstellar 
orbital motion such as could become perceptible to our 
micrometric instruments during the comparatively short 
period since precise observations were commenced by ter- 
restrial man (cf. p. 322). 

Closely related to the clusters are the nebulae (p. 3). 
Indeed, certain clusters, like the Pleiades, are so completely 
interwoven with clouds of nebulous matter, and with 
nebulous lines connecting the several stars, that one is al- 
most inclined to regard them as nebulae partly condensed into 
stellar nuclei. But the true nebulae are undoubtedly gaseous : 
spectroscopic evidence on this point is conclusive (pp. 4, 283). 

" Planetary nebulae" are a class of nearly circular light- 
clouds possessing almost planetary central disks. Their 

352 



STARSHINE 

spectra contain certain lines belonging to nebulae only, 
and ascribed to incandescence of a hypothetical substance 
"nebulium." At their centers, nebulae of this class have 
at times nuclei that look like stars. One is tempted to 
imagine them in the last stage of nebular development, and 
on the verge of becoming starry. There exist also a few 
ring-formed nebulae. 

But the type-form of nebula is the spiral nebula (cf. 
Plate 2, p. 4). Keeler thought there are 120,000 objects 
of this form within the photographic range of his big reflect- 
ing telescope at the Lick observatory. His observations 
opened the eyes of astronomers to the probability that the 
Laplacian plan of cosmic evolution may not be the one 
generally active; that a single sun like ours is less likely 
to occur in space than some more complicated system of 
suns, resulting perhaps from a great apparently whirling 
complex by us seen as a spiral nebula. 

The Andromeda nebula is by far the largest of the spirals : 
for modern long-exposure photographs have proved its spiral 
character, though it was always supposed to be elliptical 
or ring-formed until celestial photography came into general 
use. We cannot measure its distance from us by any of 
our parallax methods : but it is possible to fix for its parallax 
a limit of 0."01. The parallax cannot be much larger than 
this, or traces of it would reveal themselves to our instru- 
ments. But assuming this parallax, and the known angular 
diameter of the nebula (one and a half degrees of arc), it 
must have a linear diameter 540,000 times as great as the 
distance between the earth and the sun. 1 

It is furthermore of interest to compare this nebula's 
possible attractive force with that exerted by the sun on 

1 Note 44, Appendix. 
2 a 353 



ASTRONOMY 

the earth. It may be computed that if the nebular density- 
is only soo ofo"o o o" °f tne sun ' s > the nebula will attract the 
earth as much as the sun does. 1 So the fact that we find 
no perturbative attraction whatever in the solar system 
resulting from the nebulae, proves that, though enormous 
in size, they are of an almost inconceivable tenuity ; in fact, 
almost without any density whatever. 

To the foregoing very recent researches must be added an 
observation of most ancient date, but one that all the 
modern theories have failed to explain quite fully. This is 
the Milky Way, or "galaxy," which shows itself as a band 

of small stars — star 



© 



Fig. 104. The Galaxy. 



dust — encircling 
the sky almost like 
the celestial equa- 
tor, ecliptic, etc. It 
has many starless rifts and lanes, and several "holes" ; not- 
ably the "coal-sack," situated near the south pole of the 
heavens. It contains numerous star-clusters, but few nebulae. 
One of the most interesting is shown in Plate 29, — the "North 
America" nebula, so named because of its shape. 

The galaxy, resembling a great circle of the celestial sphere, 
must of course have two nodes, or points of intersection 
with the ecliptic circle. These are near the solstices, and 
the galactic circle makes an angle of about 60° with the eclip- 
tic. According to Sir William Herschel, the stars of the galaxy 
are spread out in a thin disk, in which our sun is also situated. 
In Fig. 104, the solar system is shown at S, and the galactic 
disk is within the rectangle. Outside the rectangle, the stars 
are fewer and farther asunder. It is evident that if we look 
in the direction A or B from S, we shall look through the 

1 Note 45, Appendix. 
354 




Plate 29. The North America Nebula. 



Photo by Barnard. 



STARSHINE 

thick part of the galaxy, and see an enormous number of 
stars projected on the celestial sphere ; but if we look toward 
C, we shall see projected on the sky only a thin part of the 
disk, and the sparser stars, outside it. And the disk, of 
course, when produced outward to the celestial sphere, will 
cut out the galactic great circle. 

Actual counts of stars have been made by Herschel and 
others, to ascertain the number per square degree of sky 
surface at various angular distances from the galaxy. The 
numbers are found to vary in the following proportions : 



Angular Distance 


Relative Number op Stars 


from Galaxy 


per Square Degree 


90° 


4 


60° 


7 


30° 


18 


0° 


122 



The unexplained difficulty with HerscheFs explanation 
of the galaxy still remains. The extreme minuteness of 
the galactic stars indicates immense distance, as does also 
their lack of observable proper motions across the sky. 
But if so enormously distant, how can the galaxy constitute 
a single disk-shaped cluster or universe ? But perhaps we 
are here inventing a difficulty, because of our inability even 
to imagine the scale of the sidereal universe. 

However this may be, we may close this part of our subject 
with a definite statement that there is no evidence in the 
possession of astronomers to indicate a " central sun "around 
which all the stars are circulating in their orbits. As already 
stated, we now believe their motions more nearly resemble 
the gyrations of the molecules in a gas under the kinetic 
theory. Gravitation probably takes hold in an appreciable 
degree, only when a couple of stellar molecules happen to 
pass near each other, speaking cosmically. 

355 



CHAPTER XXI 

THE UNIVERSE ONCE MORE 

The reader may recall that we commenced our long ex- 
planation of astronomic science in the present volume with 
a chapter entitled "The Universe." Now that we at last 
approach the end, let us once again return to the beginning, 
and reexamine the evolutionary processes of the cosmic 
universe, in the light of the astronomic knowledge we have 
been able to gain. 

Cosmogony is a name given to the various theories of the 
universe and its life-history : there is no subject more entic- 
ing to the mind of man ; none in which he is more prone 
to be misled into fields of mere speculation, quite outside 
the domain of strictest logic, based on irrefragible observa- 
tional premises. 

We have already mentioned (p. 235) the Laplacian nebular 
hypothesis, with its rotating nebulous sun, forming planets 
by the successive separation of rings : it will now be proper to 
inquire a little more closely into the admissibility of La- 
place's idea. 

It will be well to begin by summarizing the known facts 
that are favorable to Laplace : 1 

1, The planetary orbits all lie nearly in the same plane; 
and the direction of orbital motion is the same for all planets, 
and for the sun's axial rotation. 

2. The orbits are all nearly circular. 

1 Laplace, "Exposition du Systeme du Monde," p. 470, in Oeuvres de 
Laplace, Paris, MDCCCXLVL 

356 



THE UNIVERSE ONCE MORE 

3. With the exception of Uranus and Neptune, the equa- 
torial planes of the planets, and even the orbital planes 
of their satellites, all coincide approximately with the 
fundamental plane of all the orbits ; and the direction of 
the satellites' orbital motion is in general also the direction 
of the planets' axial rotation. 

4. We have accepted the Helmholtz theory (p. 294), 
that the sun's source of heat must be sought in the contrac- 
tion of its bulk : in that case the sun must have once been 
incomparably larger than at present, just as the nebular 
hypothesis requires. Laplace, of course, was not in posses- 
sion of Helmholtz' calculations when his own theory was 
published. 

Now, as a matter of logic, a correct theory must explain 
every observed fact within its range. A single contrary 
observation may destroy logically an entire theory, no 
matter how many other observations seem to confirm it. Let 
us then next enumerate some of the objections that have 
been advanced against the nebular hypothesis. 

1. Phobos, the inner satellite of Mars (p. 222), and the 
inner edge of Saturn's ring (p. 245), revolve in their orbits 
faster than the axial rotation periods of Mars and Saturn 
respectively. This will not do : the contracting planet 
should rotate faster than any satellite revolves around it, 
just as the inner planets have shorter sidereal periods than 
the outer ones. 

2. The kinetic theory of gases would lead us to expect 
(pp. 167, 222) that a gas like hydrogen would all be lost into 
space from each planet in the form of molecules, soon after 
the ring was thrown off from the sun, on account of the very 
small gravitational pull of the ring. Yet we still find hy- 
drogen in plenty on the earth. 

357 



ASTRONOMY 

3. The throwing off of the rings is in itself an hypothesis 
difficult of acceptance, on account of the presumably 
extreme rarity of the outer parts of the solar nebula, and 
consequent lack of cohesion. And why was the process of 
expelling rings intermittent instead of continuous ? 

4. The mechanical movements in the system present diffi- 
culties. For instance, the total quantity of rotary momen- 
tum now belonging to Jupiter is about -j 9 ^ of the total belong- 
ing to the entire solar system included within the orbit 
of Saturn. Yet Jupiter's mass is only about one-thousandth 
of the total mass remaining within Saturn's orbit, including 
the sun. Are we then to suppose that the present sun, at 
a single moment, parted with so small a fraction of its mass, 
which yet carried away almost all its rotary power ? 

Chamberlin and Moulton have endeavored recently to 
substitute a new and different theory for that of Laplace. 
They call it the " planetesimal hypothesis" ; and they think 
the recent evolution of stellar systems, and of our solar system, 
began with the accidental close approach (not collision) of 
two stars. If we imagine such an approach to have taken 
place, we must suppose the two bodies revolving for a time 
in orbits around their common center of gravity. If the 
approach was not very close, these orbits would be open 
curves, like the orbits of most comets when they approach 
the sun (p. 312). The two bodies would separate after a 
certain time, and would never again pass near each other. 

But while they were together, the gravitational effect of 
each on the other would be tremendous. Doubtless each 
would eject masses of highly heated gaseous matter, much as 
the great red hydrogen prominences (p. 293) are ejected from 
our sun. Upon these ejected gases, and upon the other outer 
gaseous envelopes of the two stars, gigantic tidal forces 

358 




Spiral Nebula, seen edgewise. 



Photo by Keeler. 




Owl Nebula. 
PLATE 30. Nebulae. 



Photo by Hate. 




THE UNIVERSE ONCE MORE 

would be exerted. Consequent gigantic tidal effects would 
be produced in each star by the other. Like the tidal 
crests caused by the moon in terrestrial oceans (p. 252), 
the ejected masses would travel outwards from each star, 
directly toward the other star, and directly away from it. 

Afterwards, these masses would move in strange orbits 
under the combined gravitational attractions of the two 
stars : we can in a way trace out these orbits by com- 
putational methods, which have been tried by Moulton. 
Chamberlin calls these masses " planetesimals " ; and it 
is assumed that they make their appearance in great num- 
bers and at short intervals. 
Figure 105 shows the probable 
orbits they would pursue. The 
dotted lines indicate various 
orbits ; the full lines show the 

. , , . . Fjg. 105. Planetesimal Hypothesis. 

points reached at a given in- 
stant of time by the several planetesimals pursuing the 
dotted orbits. 

When we look at the result of such a performance, what 
shall we expect to see? If we assume the instant of time 
when we make our observation to be that instant when the 
several planetesimals have just reached the full lines, we 
shall not see them traveling along their dotted orbits, but 
we shall see them all lying on the full lines. In other words, 
we shall see a spiral nebula. 

Now whatever strength there may be in this hypothesis, 
there can be no doubt that Keeler's observations of nebulae 
(p. 353) establish the fact that the spiral is the normal type 
of nebula. For comparison with Plate 2, p. 4, we give in 
Plate 30 a photograph of a nebula which is doubtless another 
spiral, but seen edgewise. The lower part of the plate 

359 



ASTRONOMY 

shows the "owl" nebula in Ursa Major, which looks more 
like a Laplacian nebulous sphere. Plate 31, p. 360, the 
"trifid" nebula, is a good example of quite irregular shape. 

So the new theory gives a notion as to the possibility of 
spirals resulting from a close approach of two stars, which 
may have been previously moving about in space aimlessly, 
after the fashion of the molecules belonging to a rarefied 
gas (p. 347). Thus the theory goes back to a very early 
stage of cosmic development, and shows how stars, even 
dark ones, can be transformed into nebulae. 

But how can the spiral nebula, in turn, develop into a sun 
and planets such as we have in our solar system? Of course, 
the sun is simply the remaining material from one of the 
original stars after the other had left it. If the approach 
was near enough to lead to a permanent orbital proximity, 
we might perhaps expect a binary system. And the sun 
provided, it is easy to imagine the origin of the planets. 
They are unusually large occasional plane tesimals, increased 
gradually by the accretion of other smaller ones, swept up 
by them as they moved along in their dotted orbits. When 
there was no large dominating planetesimal for a long time, 
a group of minor planets might result. The satellites 
must be regarded as little planetesimals that were shot out 
near the larger ones that became planets. 

Chamberlin and Moulton have traced out in detail the 
bearing of their theory upon the various objections that 
have been enumerated against the Laplacian idea, and with 
considerable success. The great advantage of their hypoth- 
esis is that it gives us an origin antedating the nebular 
stage ; that it makes a cycle of cosmic life and death ; and 
especially a cycle in accord with the actual visible appearance 
of existing nebulae. This the Laplacian theory does not do. 

360 




Photo by Hale. 



PLATE 31. The Trifid Nebula. 



THE UNIVERSE ONCE MORE 

The future of the solar system seems fairly clear under 
either hypothesis. The present state of affairs is one of 
apparently stable equilibrium ; and should continue, unless 
an accident arrives from outside the system. But even 
without such accident, the solar system cannot be eternal, 
because the gradual shrinkage of the sun cannot continue 
forever. When the time comes for the sun to lose its heat- 
radiating power, the solar system must become cold and 
dead. If, after countless ages, it shall ever thereafter be 
revivified, the cause will be a fresh approach to some 
other star, dark or brilliant, whose vast disturbing attraction 
will once more break up the solar matter into a mist : and 
if a great part of the energy remaining in the system shall 
be transformed into heat, then that mist will once again be a 
fire-mist, which may once more pass through all the stages 
of cosmic life and death. 



361 



APPENDIX 

The following notes contain explanations omitted in the text, 
and requiring occasionally a knowledge of elementary algebra, 
geometry, and trigonometry as far as the solution of plane right 
triangles. 

Note i. Declination and Right-ascension (p. 35). 

Declination corresponds exactly to latitude on the earth ; the 
declination of a star is its angular distance on the celestial sphere 
measured north or south from the celestial equator. This angular 
declination, like all angles, must, of course, be measured on some 
circle ; for measuring it we must imagine a circle drawn upon the 
sky from the star to the equator, and perpendicular to the equator. 
Such a circle, drawn for the purpose of measuring declination, 
is called an Hour-circle. The point where the hour-circle meets 
the celestial equator may be called the foot of the hour-circle. The 
right-ascension of a star is then denned as the angular distance, 
measured on the celestial equator, from the vernal equinox point 
on the equator to the foot of the hour-circle drawn from the star 
to the equator. 

Note 2. Hour-angle, etc. (p. 37). 

We may now define also the term " hour-angle," which is closely 
related to the hour-circle used in measuring right-ascension. The 
hour-angle of a star is defined as the angular distance, measured 
like right-ascension on the celestial equator, from the meridian to 
the foot of the hour-circle drawn from the star to the equator. 
Thus hour-angle and right-ascension are both arcs measured on 
the equator ; both arcs have one end in common, the foot of the 
hour-circle; but the other ends are different, being respectively 
the meridian and the vernal equinox. 

All the astronomical terms so far defined are exhibited in Fig. 
106. It represents half the celestial sphere, the half which is 

363 



ASTRONOMY 



above the horizon, and therefore visible to us. The large circle 
NESW represents the horizon; and the celestial hemisphere is 
shown projected down upon the plane of this horizon. The zenith, 
or point directly overhead, of course projects down into the center 
of the horizon circle. The great meridian circle appears as the line 

NPZS, since it must 
pass through the 
zenith Z as well as 
the north and south 
points of the horizon 
shown at N and S. 
The celestial north 
pole, which is, by 
definition, in the ce- 
lestial meridian, will 
project down to 
some point P. The 
celestial equator, 
everywhere 90° dis- 
tant from the pole 
P, will project into 
the circle WME. 




Fig. 106. The Celestial Sphere. 



Any star selected at random may be supposed to be projected 
down at the point S'. Then S'D, an arc drawn on the sphere 
through the star and perpendicular to the equator, is by definition 
an hour-circle. It is evident that all hour-circles must pass through 
the pole P. The arc DM on the celestial equator, included be- 
tween the meridian at M and the foot of the hour-circle at D, is 
the hour-angle of the star. The arc S'A is the star's altitude, or 
angular elevation above the horizon. Finally, if we draw a 
short piece of the projected ecliptic circle, we may take V to be 
one of its points of intersection with the celestial equator WME, 
the other point of intersection being of course below the horizon. 
And if we let V be that one of the two points of intersection which 
we have called the vernal equinox, then the right-ascension of the 
star S' is the arc VD, measured on the equator, and included 

364 



APPENDIX 



between the vernal equinox V and the foot of the hour-circle at D. 
The arc DS f is the declination. 

The above astronomical terms may be divided into two classes ; 
viz. : those that retain a constant position among the stars on the 
celestial sphere, and those that are as constantly shifting their 
positions among the stars on account of the daily seeming rotation 
of the whole sphere. Thus, for instance, the zenith, which is the 
point directly overhead, does not partake of the seeming turning 
of the sphere. The following little table shows the two classes of 
terms : 



Changing Positions among the Stars. 
Do not Rotate with Sphere 

Zenith 

Horizon 

Altitude 

Hour-angle 

Meridian 



Unchanging Positions among the Stars. 
Rotate with the Sphere 

Celestial Poles, 
Celestial Equator 
Ecliptic Circle 
All hour-circles 
Right-ascension 
Declination 
The stars, sun, etc. 
Vernal Equinox 

Note 3. Position of Celestial Pole (p. 40). 

The relative positions of the celestial pole and the horizon may 
be made clear by means of a simple diagram. Figure 107 shows a 
portion of the earth, with its 
center at C, and the observer 
on the surface at 0. The 
outer concentric circle HPZE 
is the celestial meridian, on 
the celestial sphere. The 
zenith Z will be directly over 
the observer at 0, on the 
prolongation of the observer's 
terrestrial radius CO. The 

celestial pole must be at some point of the celestial meridian, by 
definition. Let this point be P. The celestial equator will meet 
the meridian at E, 90° distant from P. The terrestrial pole will be 
at P', and E' will be a point of the terrestrial equator. The angle 
E'CO will be the terrestrial latitude of the point 0, since it is the 

365 




Position of Celestial Pole. 



ASTRONOMY 

angular distance of the point from the terrestrial equator at E '. 
The angle PCH is the altitude or angular elevation of the celestial 
pole above the horizon at H., For H, as we know, is the north 
point of the horizon for an observer at 0. 

But %ZCE = %PCH, since PC is perpendicular to CE, and 
HC perpendicular to ZC. Hence we have a demonstration that 
the altitude of the celestial pole is everywhere equal to the terres- 
trial latitude of the observer. Thus, as stated in the text, this alti- 
tude will be 90° to an observer at the pole of the earth, where 
the latitude is 90° ; and it will be 0° to an observer at the equator, 
where the terrestrial latitude is likewise 0°. 

Note 4. Stars that never Set (p. 43). 

It is evident that these stars are the ones whose diurnal circles 
have an angular distance from the celestial pole less than PH 
(Fig. 107) ; i.e. less than the observer's terrestrial latitude, 
These stars will have a declination greater than (90° — latitude). 

Note 5. Sidereal Time (p. 67). 

The sidereal time, or hour-angle of the vernal equinox, is the 
arc VM in Fig. 106. 

To make the definition of sidereal time perfectly general, as- 
tronomers count all hour-angles westward from the meridian, and 
allow them to increase continuously to 24 h . Thus, an hour- 
angle l h east from the meridian, corresponding to 23 h sidereal 
time, would be called a west hour-angle of 23 h . 

Note 6. Right-ascension of the Meridian (p. 67). 

Again recurring to Fig. 106, it is clear from our definitions that 
the right-ascension of a star on the meridian is the arc VM ; and 
we have seen in Note 5 that this identical arc is also the sidereal 
time. Therefore the sidereal time and the right-ascension of the 
meridian at any instant are the same. 

The general relation of hour-angle, right-ascension, and sidereal 
time may also be deduced from Fig. 106. We have from our 
definitions : 

VD = right-ascension of star S', 
DM = hour-angle of star S', 
VM = sidereal time. 
366 



APPENDIX 



And since, from Fig. 106 : 

VM = VD + DM, 
it follows that in general : 

Sidereal time = right-ascension + hour-angle. 
Hour-angle = sidereal time — right-ascension. 
The last equation enables us to ascertain the hour-angle of a 
star at any instant, if we know its right-ascension, and have a 
correct sidereal clock at hand. 

Note 7. Hour-angle of Visible Sun (p. 68). 

In Fig. 106, if we let S' be the visible sun at any instant, its 
hour-angle is the arc DM, measured in hours, minutes, and seconds. 
This same arc is also the apparent solar time at that instant. 

Note 8. Terrestrial and Celestial Meridians (p. 73). 

If we imagine a line drawn from the center of the earth to the 
observer, and thence continued outward to the celestial sphere, 
it will pierce the 
sphere at the ob- 
server's zenith. The 
terrestrial meridian, 
by definition, passes 
through the north 
pole of the earth and 
the observer. The 
celestial meridian, 
also by definition, 
passes through the 
celestial north pole 
and the observer's 
zenith. Therefore 
the celestial merid- 
ian is a projection 
of the terrestrial 
meridian outward 
on the celestial sphere. Figure 108 is like Fig. 106, with the 
addition of a second celestial meridian. The figure represents the 
celestial sphere projected down upon the horizon of New York, 

367 



/ p 

/ / \ z ' 

w ^ / I \ __ 

\. ft \ 

\ B m 



Fig. 108. Time Differences. 



ASTRONOMY 

of which the zenith appears as before at Z. The projection of 
the zenith of Greenwich at the same instant is at Z' . Therefore 
PZ'M' will be the projection of part of the celestial meridian 
of Greenwich. The sun and vernal equinox are projected at 
S' and V'j as before. Then DM is the sun's hour-angle at New 
York; DM', its hour-angle at the same instant at Greenwich. 
MM', which measures the angle between the two celestial merid- 
ians, is also the difference of the two hour-angles, or the solar 
time difference between New York and Greenwich. And this is 
the same as the longitude difference, measured by the two cor- 
responding terrestrial meridians on the earth inside the celestial 
sphere. 

At the same moment, VM and VM' are the hour-angles of the 
vernal equinox at New York and Greenwich ; and MM' is also 
the sidereal time difference. Consequently, the sidereal and solar 
time differences are equal and identical ; they are both measured 
by the same arc MM '. 

Note 9. Angle of Gnomon (p. 79). 

It is evident that the ''factor" in the table is simply the tangent 
of the latitude. In Fig. 24, 

be = ac tan bac, 
and if the tabular factor is tan latitude, the construction of the 
figure will make the angle bac equal to the latitude, as required 
for the gnomon. 

Note 10. Mathematical Principles of the Sundial (pp. 80, 84). 

To demonstrate the correctness of the construction given in the 
text for drawing a sundial, it is necessary to have recourse to 
the well-known formulas of spherical trigonometry relating to the 
solution of right-angled spherical triangles. The accompanying 
Fig. 109 represents the conditions of the problem. The large 
circle ZPNQS is the celestial meridian. The circle NIVS is the 
horizon, on the plane of which the dial is to be drawn. The center 
of the dial is at ; and QP is the axis of the celestial sphere. As 
the edge of the gnomon is parallel to the axis QP, we may regard 
it as lying in that axis, because the sun will appear to rotate around 
the edge of the gnomon (p. 84). So we may consider the edge of 

368 



APPENDIX 




the gnomon to start at 0, and to extend a short distance in the 
direction OP. 

Now suppose OIV, situated in the horizon plane NIVS, to be 
the direction in which the shadow falls at four o'clock. Then, 
remembering that solar time is simply the hour-angle of the sun, 
we recall that "four 
o'clock" means that the 
sun's hour-angle is four 
hours, or 60°. We may 
suppose the sun to appear 
at the point S' at four 
o'clock. Then, from the 
definition of hour-angle 
(p. 363), the sun is then 
distant 60° from the 
meridian; or the angle 
ZPS' is 60°. The op- 
posite angle NPIV, be- 
ing equal to ZPS', is thus 
also 60°. 

Now let us consider the spherical triangle formed on the celestial 
sphere by the three points P, N, and IV. In it we know the side 
PN, for it is the elevation of the celestial pole above the horizon, 
and therefore equal to the latitude of the place where the dial is 
to be used (p. 365). As we have just seen, we also know the angle 
NPIV, which is 60°. And we know the angle PNIV to be a 
right angle, because the celestial meridian is perpendicular to 
the horizon. 

According to the principles of spherical trigonometry, if we 
know one side and one acute angle of a right-angled spherical 
triangle, we can calculate all the other parts of the triangle. In 
the present problem, we need only calculate the side NIV. For 
this measures the angle NOIV, which is the dial angle for the 
four-o'clock line, or the angle which the four-o'clock line makes 
with the north-and-south line ON. 

In the same way, we can calculate the dial angles for the one- 
o'clock, three-o'clock lines, etc. The twelve-o'clock line, or noon- 
2 b 369 



Fig. 109. The Sundial. 



ASTRONOMY 

line, is of course ON ; for at noon the sun is on the meridian, and 
the shadow of a gnomon pointing at the celestial pole will then fall 
due north. We might construct the dial by simply laying off the 
proper computed angles for the various hours from the dial 
center 0. 

The trigonometric formula for calculating the side NIV, or 
the dial angle NO IV, is : 

tan NIV = tan NPIV sin PN. 

And if we let : 

u = dial angle for any hour, 
t = corresponding hour-angle of the sun, 
I = latitude of the place, 
then the general formula is : 

tan u = tan t sin I. 

The dial angles calculated by this formula for the latitude of 
New York are as follows : 

XII. 0° 0' 

I. 9° 56' 

II. 20° 40' 

III. 33° 10' 

IV. 48° 32' 

V. 67° 42' 

VI. 90° 0' 

It now remains to show that the construction given in the text 
(Fig. 25) is in accord with the above general formula. In this 
figure we have really drawn two half -dials, so as to allow for the 
thickness of the gnomon. To prove the construction of Fig. 25 
correct, we have now to show, for instance, that : 

tan cal = tan 15° sin I. 

The factors given in the table on p. 80 are the sines of the 
latitudes I. Therefore, since we made Mc (Fig. 25) equal to ca 
multiplied by the factor in the table, it follows that : 

Mc = ca sin I. (1) 

We made the angle cMI (Fig. 25) equal to one-sixth of a right 

370 



APPENDIX 

angle, or 15°. Therefore, from the right-angled plane triangle 
Mel, we have : 

Ic 
TT~— tan 15°, 

Mc ' 

or: 



Ic = Mc tan 15°. 




Substituting the value of Mc from equation (1) gives : 




Ic = ca tan 15° sin I, 
or : 




Ic 

— = tan 15° sin I. 

ca 


(2) 


Now from the right-angled plane triangle cla t we have : 




— = tan cal. 
ca 


(3) 



Substituting from equation (2) in equation (3), we have 

tan cal = tan 15° sin I ; 
and the correctness of the construction in Fig. 25 is proved, since 
the above equation accords with the general form : 
tan u = tan t sin l. 

Note ii. Theory of Foucault Experiment (p. 91). 

We have explained the conditions of the problem if the experi- 
ment were performed at the north pole of the earth. There the 
point of suspension of the pendulum's wire would of course be 
situated in the prolongation of the earth's axis, and would conse- 
quently not move as a result of the earth's axial rotation, which 
is the only motion of the earth here requiring consideration. In 
any other latitude, the point of suspension would go around as 
the earth rotates : it is therefore necessary to explain further the 
statement that the direction in space of the pendulum's plane of 
oscillation tends to remain constant. The fact is that when the 
point of suspension moves, the plane of oscillation moves also ; 
but it tends to occupy a position constantly parallel to itself. 
Any one can satisfy himself that this is correct by fastening a 
small metal ball to a string and letting it oscillate, the end of the 
string being held in the experimenter's hand. It will be found 
that the experimenter may walk across the room, carrying the end 

371 



ASTRONOMY 




of the string ; yet the plane of oscillation will remain constantly 

parallel to itself. 

So much being premised, we may now proceed to calculate the 

rate at which the marks under the pendulum should rotate. Let us 
suppose we start the pendulum swinging in 
a north-and-south direction, and therefore 
directly under the celestial meridian, and in 
the plane of the meridian. In Fig. 110, let 
NABCS be the meridian directly over the 
pendulum when we start it swinging, and 
suppose it swings between two points in the 
room corresponding to the points A and B 
of this meridian. In a second of time the 
earth's rotation will have brought a new 
celestial meridian over the swinging pendu- 
lum, and the old one will have gone to the 
position NA'B'C'S. 

But the pendulum will still swing parallel 
to the plane of the first meridian, and the 
rotation shown by the experiment will be 
equal to the angle between the two meridians. 

Let us draw Fig. Ill to show this angle. This figure is like a 

map in a geography book. If the original meridian was AB, and 

the meridian at the end of one second A'B', the rotation shown 

by the pendulum will be the angle between these two lines. If 

we draw A'M perpendicular to BB' , the rotation 

angle will be the angle MAB' . Let us call this 

angle a. 

It is well known that on a map of the earth's 

equatorial regions the terrestrial meridians are 

practically parallel : there is no "convergence of 

meridians" there ; and there would be no Fou- 

cault effect. Near the pole the angle between 

the meridians is a maximum : there the Foucault 

effect is also greatest. 

In this way we translate our astronomical problem into terms of 

geometry : it is now merely a question of simple geometry to as- 

372 



5 
Fig. 110. The Foucault 
Experiment. 



B'M B 

Fig. 111. The Fou- 
cault Experiment. 



APPENDIX 

certain the angle of convergence between two neighboring meridians 
on the earth in any latitude, such as that of New York, for instance ; 
and this angle will be the Foucault pendulum rate of rotation. 

We see at once from Fig. Ill that, in any latitude, we have 
from the triangle A'B'M: 

MB' 



tana 



MA" 



and since for very small angles like a the tangent and the arc are 
equal, we may write : 

a = **L (1) 

MA' K J 

Referring again to Fig. 110, which we may now take to represent 
the earth instead of the celestial sphere, we observe that the latitude 
arcs AA', BB', and CC are all arcs of circles whose radii are 
PA', P'B', and OC. The last radius OC is the earth's radius, 
because we shall consider CC to be an arc of the equator. Now 
suppose the point B' to correspond to the terrestrial latitude I'. 
Then V is the angle B'OC, for the latitude is the angular distance 
of B' from the equator. But the line P'B' in the plane of the circle 
NA'B'S is proportional to the cosine of the angle B'OC. Simi- 
larly, the radii of all arcs like AA', BB', etc., are simply proportional 
to the cosines of the latitudes corresponding to the points A', B' } 
etc. 

But the arcs themselves must be proportional to their radii. 
So it follows that the linear lengths of the arc AA', BB', are also 
proportional to the cosines of the corresponding latitudes. 

We have called V the latitude corresponding to the point B' . 
Let us call I the latitude corresponding to A'. Now we have found 
that the arcs A A' and BB' are proportional in length to the 
cosines of the latitudes I and V. Therefore the difference between 
A A' and BB' must be proportional to the difference of the same 
cosines, which we may express by the following equation, in which 
K is simply a constant denoting proportionality : 

BB' - AA' = K (cos V - cos I). 

But, from Fig. Ill: 

MB' = BB' - AA'. 
373 



ASTRONOMY 

Consequently, from the preceding equation : 

MB' = K (cos V - cos I). (2) 

Now, in Fig. 110, draw the line A'Q perpendicular to P'B', com- 
pleting a little right-angled triangle A'B'Q. (We may regard the 
short arc A'B' as here equivalent to a straight line.) Then we 
have: 

QB' = cos V — cos I, 

and : QB' = A'B' sin QA'B'. 

But: QA'B' = B'OC = V \ 

therefore : QB' = A'B' sin /'. 

But: A'B' = (I - I'). 

Consequently : QB' = cos V — cos I = (I — V) sin V. 

It then follows from equation (2) that : 

MB' = K(l - V) sin V. (3) 

We also have, obviously, from Fig. Ill : 

MA' =1-1'. (4) 

Now substituting from equations (3) and (4) in equation (1), we 
have finally : 

a = K sin V. (5) 

This simple equation (5) establishes the important principle 
that the rate of rotation of the Foucault pendulum in one second 
must everywhere be proportional to the sine of the latitude of the 
place where the experiment is performed. 

It is further obviously indifferent whether the original impulse 
was given to the pendulum in the direction of the meridian ; for 
whatever angle the original impulse made with the original merid- 
ian, at the end of one second of time that angle will have changed 
by the same quantity a with respect to the meridian. 

It is quite easy to find the value of the constant K in equation 
(5). For at the north pole, sin V = 1, since V = 90°. Therefore, 
at the pole, equation (5) becomes : 

a = K. 

But we already know that at the pole the pendulum must 
make one complete revolution of 360° in 24 hours. So it must 

374 



APPENDIX 

there revolve at the rate of 15° per hour, or 15' per minute. With 
this value of K we therefore have in any latitude V : 

Rate of revolution = (15' per minute) sin V. 

In New York, for instance, 

V = 40° 48', sin V = 0.65. 
Rate of rotation = 9/75 per minute. 

The above demonstration of the Foucault pendulum theory is 
not rigorous, but it is sufficiently accurate for ordinary purposes, 
provided the duration of the experiment is not much greater than 
one hour. 

Note 12. The Torsion Constant (p. 108). 

The problem of ascertaining the torsion constant T from the 
time of oscillation of the torsion balance is quite analogous to the 
corresponding problem of determining the length of an ordinary 
pendulum from its time of vibration. It is shown in books on 
physics that if we let : 

t = vibration time of an ordinary simple pendulum, 
I = length of the pendulum, 
g = the force of gravity on the earth, 
tr = the ratio 3.1416, 
then: 

•-* 

An analogous formula exists in the case of the torsion balance, 
except that instead of g, the force of gravity, the formula involves 
T, the torsion constant. We now let I represent the entire length 
ab of the torsion balance arm (Figs. 32 and 33), andm the mass of 
either small ball a or b. Then the torsion balance formula is : 

and those readers who are acquainted with the science of mechanics 
will note that 2ml -j is the " moment of inertia" of the entire 
balance. 

375 



ASTRONOMY 

Solving this equation for T gives : 

rp _ 7T 2 ml 2 . 

2t 2 ' 
and this equation will make known the value of T for any torsion 
balance after we have observed its vibration time t, measured 
the length of the arm ab, and ascertained m by weighing the small 
balls in an ordinary balance. 

Note 13. The Cavendish Experiment (p. 110). 

Returning now to Fig. 33, let us use the following notation : 
M = mass in grams of either big lead ball, 
m = mass in grams of either small ball, 
d = measured distance in centimeters from the position of rest b' 

to B', the center of the big lead ball, 
g = the acceleration due to the " force of gravity," as used in 

physics, or 981 centimeters, 
I = length of torsion balance arm, or the distance ab, in centi- 
meters, 
/ = the force with which both big balls turn the balance. 
Now, according to Newton's law, the attractive force between 
the balls B' and b' is (p. 103) : 

r Mm n x 

G ir> (1) 

in which formula G is a so-called " gravitational constant," in- 
troduced to indicate that the attraction is proportional to —-=r-> 

a 1 

not equal to it. 

The distance from B' to a' is : 



Vd 2 +Z 2 ; 
consequently, the attractive force existing between B f and a' is : 

0^-, (2) 

Both forces (1) and (2) tend to turn the torsion balance. They 
act against each other, however, tending to rotate the balance 
in opposite directions. And the force (1) is larger than (2) ; 
so that it will determine the final direction of rotation. 

376 



APPENDIX 

Furthermore, the entire force (1) tends to turn the balance, 
while only a small part or " component" of (2) has such an effect. 
We can easily find this component, which acts from B' upon a' so 
as to turn the balance. According to the so-called " parallelogram 
of forces" this component is : 

°^r P smBab > 

or: 

Mm d 



G 



or, finally: 

G Mmd . - (3) 

(d 2 + vy 

The effective force tending to rotate the balance, and resulting 
from the big ball B', will be the difference of (1) and (3). It will 
be J/, since, in our notation, / is the force with which both big balls 
tend to rotate the balance. By subtracting (3) from (1) we thus 
obtain the equation : 

y = GMmf} 2 i~\ (4) 

\d 2 ( d 2 + pyj 



For brevity, let us put : 
1 
d2 (d 2 + P) 



^=4 ^-i- (5) 



Then we have : 

J/ = GMmD, (6) 

and, solving for M, we obtain : 

M=i.-^-. (7) 

G 2mD 

The force / must be determined from observations of the torsion 
balance, when under the influence of the big lead balls. Trans- 
ferring these big balls from the position A' , B' to the position A", 
B" usually rotates the balance through a very small angle only. 
It is therefore necessary to measure this angle by very delicate 
means. For this purpose a small light mirror is attached to the 
center of the arm ab of the balance, and rotation is measured by 
allowing a strong beam of light to fall on this mirror, and to be 

377 



ASTRONOMY 

thence reflected upon a scale at some distance from the apparatus. 
The rotation of the balance is thus magnified, and can be measured 
without difficulty. 

To introduce these measures into our formulas, let : 
a = the total change in centimeters of the light on the scale brought 

about by changing the big balls from A', B' to A", B" . 
Q = the distance of the scale from the mirror. 

To employ the units usual in measures of this kind, we must 
reduce the motion a to what it would have been on a scale at 

unit distance from the mirror. This would be — • We must also 

allow for the well-known fact that a moving reflected beam changes 
its direction twice as fast as the mirror turns. This reduces the 

motion on the scale at unit distance to — — • Finally, we must 

Z \£ 

again divide by 2 to obtain the effect corresponding to the half 
motion W, instead of the whole motion b"b f , since we are calcu- 
lating the disturbance of the balance from a position of rest, and 
have measured its motion between two positions of extreme 
disturbance. This gives the observed motion on the scale, to be 
used in the further calculations, as : 

— • (8) 

Now it is a principle underlying the torsion or twisting of rods 
or fibers, a principle verified easily by experiment, that the force 
required to twist the rod or fiber through any angle is proportional 
to that angle. For instance, if a certain force would turn the 
torsion balance through an angle of 10°, it would require just 
twice as much force to turn it through 20°. It follows from 
this principle, and from the definition of the torsional constant T, 
that the force required to turn the balance through the angle (8) 
is: 

IQ T > (9) 

where readers familiar with Mechanics will note that T is really 
the "turning moment" for unit angle applied at unit distance 
from the center. 

378 



APPENDIX 

This expression (9) is not yet equal to the force /, because / is 
applied at the ends of the balance arms where the small balls are. 

The length of this balance arm being -, we see that (9) must be 
equal to / - ; and so we may write the equation : 

From this we have : 

/(observed) -lg. (11) 

We have already obtained in Note 12 (p. 375) an expression for 
T as follows : 

T = ^f • (12) 

With the help of equations (11) and (12) we can compute the 
observed force / from our observations of a and Q, and the known 
dimensions, etc., of the parts of the balance. 

Next we can establish easily an expression for the attractive 
force existing between either little ball and the earth. For this 
purpose, let 

R = the radius of the earth, in centimeters, 
E = mass of the earth, in grams. 
Then we have : 

Attractive force between small ball and earth = G-=—* (13) 

R 

Equation (13) follows directly from Newton's law, if we recall 
that the earth attracts bodies exterior to it precisely as if the entire 
mass of the earth were concentrated at its center. Thus the radius 
of the earth becomes the distance between the earth and the 
small ball, and its square appears in the denominator of equa- 
tion (13). 

Furthermore, according to the teaching of Physics, the attractive 
force existing between the small ball and the earth is also measured 
by the weight of the small ball, since weight is merely the result 
of such attractive force of the earth. And in physics, the weight 

379 



ASTRONOMY 

of any object is shown to be equal to its mass multiplied by the 
force of gravity, g. So we have : 

Attractive force between small ball and earth = mg. (14) 
Equating the right-hand members of equations (13) and (14) 



or: 



VCJ 


) . 


mg ■■ 


n Em 




(15) 






E = 


.R 2 g 
G 




(16) 


If 


we now 


divide equation (16) by equation (7), 


we obtain 








E 
M' 


= j ■ mW 9 D. 




(17) 



We now obtain the value of / from equation (11) by the help of 
equation (12). This gives : 

f ~1¥Q' (18) 

Substituting from equation (18) in equation (17) gives, finally : 

g-gfSgP.fl.lf. (19) 

7r 2 l a 

Equation (19) enables us to calculate the mass of the earth, E, in 
terms of the mass of either big lead ball, M . It will be noted that 
the only quantities used in equation (19) and actually observed in 
the Cavendish experiment are a and Q. Most of the other quanti- 
ties are ascertained by measurements and weighings before the tor- 
sion balance is put together. The time of vibration, t, is found in 
seconds by observing the combined duration of a considerable num- 
ber of oscillations, made with the big lead balls entirely removed. 
In the actual apparatus mounted for use in the astronomical 
lecture-room at Columbia University, New York, the following 
numerical data exist : 

t — 281.5 seconds, 
d = 5.3 centimeters, 
Z = 3.6 centimeters, 
g = 981 centimeters, 
7r = 3.1416 centimeters, 
M = 2750 grams, 
380 



APPENDIX 

and the radius of the earth is : 

R = 6.371 X 10 8 centimeters. 

With these numbers we obtain from equation (19) : 

E = 0.30 X 10 27 • 2 g ram s. 
a 

In an actual experiment the writer found : 
Q = 189 centimeters, 
a = 10.86 centimeters. 
Therefore : 

Q = 17 A, 
and : 

E = 5.22 X 10 27 grams. 

The present accepted value of the earth's mass is : 
6 X 10 27 grams ; 
so that the result of the above lecture-room experiment is fairly 
satisfactory. 

Note 14. Linear Distances from Angles Alone (p. 119). 

The simple Figure 112 shows the correctness of the principle 
stated in the text. Suppose, for instance, that the three angles 
of a triangle are given, and it is required to 
draw the triangle. It is not possible to do so ; 
because, with the given angles, we do not 
know whether we should make it of the 

a i-i • n j-i • m Fig. 112. Distance from 

size A, or the size B, or any other size. To Angles. 

know the triangle fully, we must know the 
length of at least one side. Angles alone enable us to draw a figure 
which is geometrically similar to the required figure, but they do 
not enable us to draw the figure itself to scale. 

Note 15. Calendar Rule (p. 144). 

To demonstrate this rule, we begin by assuming that our era 
commenced with a year numbered 0, so that 1913 was the 1914th 
year of the era. Of course there was not really an initial year 0, 
but we can imagine the calendar extended to that time. Then the 
principle on which our rule is based consists in calculating the 
number of days from January 1 of the year to the date under in- 

381 




ASTRONOMY 

vestigation, and ascertaining how many weeks elapsed in the 
interval. 

It happens that January 1 of the year was a Sunday. Let us 
next compute the number of days between January 1 of the year 
and March 1 of any year, such as 1913. We select March 1, 
because it is desirable for the moment to use a date that follows 
the possible extra day inserted as February 29 in leap-years. 

Let us indicate the year number, such as 1913, by the letter y, 
and the century number, such as 19 in the year 1913, by the letter 
c. The total number of days from January 1 in the year 0, to 
March 1 of the same year, is 59, for the year was not a leap-year. 
Consequently, if there were no leap-years, we should have : 

No. of days from Jan. 1, year 0, to Mar. 1, year y — 365 y + 59. 

As each leap-year adds one day, we must increase this by the 
number of leap-years from the year to the year y, and including 
the year y, if it be a leap-year. To find this number, let us divide 
c and y by 4, and call the remainders after the division n and r 3 . 
Then it is clear that under the Gregorian rule the number of leap- 
years will be ; 

I (y - n) - c + i (c - n). 

Furthermore, this number will be a whole number, because we can 
prove easily that y — r 3 and c — r\ are both divisible exactly by 4, 
without remainder. 

The proof of this is as follows : If we divide any number what- 
ever, N, by some other number D, and find from the division a 
quotient Q and a remainder R ; then, if we divide N — R by D, we 
shall again find the same quotient Q, but the remainder will now 
be 0. Thus, if we divide 1913 by 4, we find the quotient Q is 478 
and the remainder R is 1. If we now subtract this remainder 1 
from the original number 1913, we have for N — R, 1912. This 
being divided by the same divisor 4, gives the old quotient Q as 
478, but the remainder is now 0. This shows that our expression 
for the number of leap-years is a whole number, as it should be. 

We then have, by addition of the number of leap-years : 

Total no. of days from Jan. 1, year 0, to Mar. 1, year y 
= 365 y + 59 + i (y - r 3 ) - c + i (c - n). 
382 



APPENDIX 

Now, if March 1 in the year y is a Sunday, like the first day of 
the era, the above number must be divisible exactly by 7. But if 
March 1 in the year y is Monday, one day later than Sunday, 
we can make the above number divisible by 7 if we subtract 1 
from it; and 1 is the week-day number for Monday, minus 1. 
Similarly, for Wednesday, for which the week-day number is 4, we 
would subtract 3. In general, let us indicate by w the week-day 
number of March 1, whatever it may be in the year y, and subtract 
w — 1 from the above total number of days. This gives : 

365 y + 59 + J (y - r s ) - c + i (c - n) - (w - 1), 

and this number is now always divisible exactly by 7. 

Our real problem is to determine (w — 1) from the fact that the 
number just obtained is thus divisible exactly by 7. In doing this 
we may evidently increase or diminish our number by any exact 
multiple of 7 without impairing its divisibility by 7 or affecting the 
value of w. We shall introduce two new remainders r 2 and r 4 , by 
dividing the century number c, and the year number y, by 7, just 
as we have already divided them both by 4. 

This having been done, we may correct our total number of 
days as follows, noting, of course, that each number added or sub- 
tracted is divisible exactly by 7. We shall add : 





i (y - n) + J(c -ri), 


subtract 


364 y + 56, 


add 


7 n + 7 r 3 , 


subtract 


3 (y - n) + (c - r 2 ), 



and so our total number becomes : 

3 + r 2 + 5 r 3 + 5 n + 3 r 4 - (w - 1). 

This number is now made up of remainders only. It will be a 
comparatively small number, as no remainder is larger than 6 ; 
and it is still divisible exactly by 7. It is therefore clear that 
(w— 1) is simply the remainder that will occur in the division by 
7 of the number : 

3 + 5 n + r 2 + 5 r 3 + 3 r 4 , 
and thus (w — 1) is determined for March 1 in the year y. 

383 



ASTRONOMY 

But we need to find (w — 1) for any day in the year y, not merely 
for March 1. To accomplish this for any other day in March, 
say the 3d, for instance, we have merely to add 2 to the above 
number, before dividing by 7, because March 3 comes two days 
later than March 1. In general, if we indicate by d the date in 
March for which the week-day is required, we must add (d — 1) 
to the above number. This gives, for March d th : 

3 + 5 n + r 2 + 5 r 3 + 3 n + (d - 1), 
or : 2 + 5 n + r 2 + 5 r 3 + 3 r 4 + d ; 

and this number being divided by 7 will give the (w — 1) of March 
d for a remainder. 

The same expression will hold for April, if we add 31, because 
there are 31 days in March. Adding 31, and deducting 28, an exact 
multiple of 7, gives for April : 

5 + 5 n + r 2 + 5 r 3 + 3 r 4 + d. 

A similar expression holds for each month, a difference occurring 
only in the number at the beginning of the expression. If we in- 
dicate that month-number by m, we may write for any month : 

m + 5 ri + r 2 + 5 r 3 + 3 r 4 + d. 

The values of m for the various months may then be written in a 
little table (see Rule, p. 144). In forming this table it is necessary 
to remember that there will be a slight difference between the 
ra's for leap-years and ordinary years. We have started with 
the formula for March 1, in order to avoid this difference as 
much as possible. After that date in the year there is no difference. 
But in January and February the leap-year m's are smaller by 
1 than those for ordinary years, on account of the interpolated 
February 29. 

The entire rule may be arranged in the accompanying tabular 
form. That part of the formula which does not vary in a whole 
century, namely, 5 n + r 2 , we have designated by K. In the Julian 
calendar K is evidently always 0, because there is no century ex- 
ception in the leap-year rule of that calendar. For the sake of 
symmetry, we have here indicated the final remainder (w — 1) 
by r 6 . 

384 



APPENDIX 

Calculation of Week-day, Gregorian or Julian Calendar 



Formula 
(d=Day of the Month) 


Table of m 






Divide 


by 


and call 

the 
remainder 




Ord'y 
Year 


Leap 
Year 




Century No. 


4 


n 


Jan. 


6 


5 


1 


Sunday 


Century No. 


7 


r 2 


Feb. 


2 


1 


2 


Monday 


Year No. 


4 


n 


March 


2 


2 


3 


Tuesday 


Year No. 


7 


n 


April 


5 


5 


4 


Wednesday 


5r 3 + 3r 4 + in 
+ m + d\ 


7 




May 








5 


Thursday 


^5 


June 


3 


3 


6 


Friday 








July 


5 


5 


7 


Saturday 








Aug. 


1 


1 












Sept. 


4 


4 












Oct. 


6 


6 












Nov. 


2 


2 












Dec. 


4 


4 







where K = 5 n + r 2 , Gregorian ; 

K = 0, Julian. 
(Gregorian K = 20, from 1900 to 1999.) 
Week-day No. = r 6 + 1. 



Note 16. Gauss' Rule for Easter (p. 148). 

To demonstrate the rule, we shall consider the Julian calendar 
first, and then modify our results to accord with the present 
Gregorian calendar. 

The lunar month of chronology, or the period of the moon's 
orbital revolution around the earth, is approximately 29J days 
long. In making the ecclesiastical calendar it was therefore 
decided to have lunar months of 29 and 30 days occur alternately 
as a general rule. But for a reason to be explained in a moment, 
an extra lunar month of 30 days is inserted at the end of every 
third year for six successive periods of three years each, or eighteen 
years in all. Then, one year later, at the end of the nineteenth 
year, an additional extra lunar month of 29 days is further in- 
serted in the calendar. 

2 c 385 



ASTRONOMY 

The lunar calendar for nineteen years then stands as follows : 

3 years (36 months) alternating 29 and 30 days, total . . 1062 days 

Extra months of 30 days 30 days 

The above repeated five times more (1092 x 5) 5460 days 

The 19th year of 12 months alternating 29 and 30 days . . 354 days 

The final extra month of 29 days 29 days 

Total 6935 days 

The above calculation takes no account of leap-years, which 
occur every fourth year in the Julian calendar. To get these 
leap-years into the lunar calendar, too, the ecclesiastical chronol- 
ogists adopted the simple plan of putting an extra day into the 
lunar month of February, whenever it is put into the civil month of 
February. In 19 years this will happen five times when any one 
of the first three years of the 19 is a leap-year ; but if the fourth 
year of the 19 is a leap-year, it will happen four times only. Thus, 
on the average : 

19 years will have 6935 + 5, or 6940 days three times, and 
19 years will have 6935 -f- 4, or 6939 days once. 

The mean of these figures is 6939f days ; and this is the average 
number of days in 19 lunar years, according to accepted chronologic 
rules. 

Now the length of a Julian tropical or calendar year is 365 J days. 
Consequently, 19 Julian years will contain 365J X 19, or 6939f , 
days, agreeing exactly with the lunar figures just found. This 
agreement is evidently not accidental, but is the result of the above 
conventional and arbitrary rules for the extra lunar months. 

One important thing follows from this agreement : if we write 
the calendar dates of full-moon for a period of 19 years, these calen- 
dar dates will then be repeated in the next and in every subsequent 
period of 19 years. Now it so happens that the year 0, or the 
year next preceding the year 1 of our era, was the first year of a 
19-year cycle. Consequently, the year 1 was the second of the 
19-year cycle, the year 2 the third, and the year 19 the first of the 
next cycle. It is clear that, in general, if we divide the year 
number y by 19, and call the remainder r 6 , then r 6 + 1 will be 
the position of the year y in a 19-year cycle. 

386 



APPENDIX 



The next step is to find for any year the date of the Easter full- 
moon, which, according to the Nicene council's decree, is the first 
to fall on March 21 or thereafter. Let us call the date of this 
full-moon March 21 + P, and suppose dates in March to be car- 
ried over into April, so that April 1 will be called March 32. Now 
it so happens that in the year preceding the year 1, the Easter 
full-moon, Julian calendar, fell on March 36, so that P was then 15. 
As there are 354 (12 X 29§) days in a lunar twelve-month, it is 
clear that in the year 1 Easter full-moon must have occurred 11 
(which is 365-354) days earlier. And in each succeeding year of 
the 19-year period, Easter full-moon must have occurred either 11 
days earlier than in the preceding year, or 19 (which is 30-11) 
days later. Of course the occasions when it occurs 19 days later 
are accounted for by the extra 30-day months inserted every three 
years. The following table exhibits the above state of affairs : 



Year 


r« 


p 










15 




1 


1 


4 


11 days earlier than year 


2 


2 


23 


19 days later than year 1 


3 


3 


12 


11 days earlier than year 2 


4 


4 


1 


11 days earlier than year 3 


5 


5 


20 


19 days later than year 4, 


etc. 






etc. 



It is clear that we shall have, in general, if we let v and x be 
two unknown whole numbers : 

P = 15 + 19 x - 11 v, 
or: 

P = 15 + 19 (x + v) - 30 v. 

It is further clear that in this equation : 

x + v = r 6 , 
because, to get P in the table above, we have always added 19 r 6 and 
then subtracted the largest possible value of 30 v. So we may 
write : 

P = 19 r 6 + 15 - 30 v. 

387 



ASTRONOMY 



From this it appears that P is simply the remainder occurring 
in the division of 19 r 6 + 15 by 30. If we call this remainder r 7 
we can therefore find the date of Easter full-moon in the Julian 
calendar thus : 





Divide 


by 


and call the 
remainder 




Year No., y, 
19 r 6 + 15 


19 
30 


7*6 


And the date of Easter full-moon, 
Julian calendar, is March 21 + r 7 . 



The above method of calculation not only applies to the first 
period of 19 years, but is entirely general ; because, as we have seen, 
subsequent 19-year periods simply repeat the same dates of full- 
moon. 

We must now pass to the Gregorian calendar. It is evident that 
the two calendars are in accord at the beginning of the era, and 
do not diverge until the year 100, when the Gregorian calendar 
omits a Julian leap-year. This will of course change P by one day, 
and this same difference of one day will continue from the year 100 
to the year 199. From 200 to 299 there will be a difference of two 
days, etc. 

It is clear that we can allow for this cause of difference between 
the two calendars by varying the number 15 that occurs in the 
quantity 19 r 6 + 15. Let us call this variable number M. Then, 
in both calendars, M is 15 from the beginning of the era to the year 
99. In the Gregorian calendar M increases by 1 each century 
thereafter, except that for every fourth century this increase is 
omitted because of the Gregorian leap-year exception. Using 
our former notation, in which c is the century number, we have for 
the Gregorian calendar : 

M = 15 + c- J (c- n). 

But this value of M, thus corrected for the Gregorian leap-year, 
is not yet quite right. A further last correction is still necessary 
on account of a slight inaccuracy in the lunar period of 19 years. 
A lunar month is not exactly 29J days long; its true length is 
29.530586 days. So the 235 lunar months of a 19-year period 

388 



APPENDIX 

really amount to 235 X 29.530588 days ; or 6939 days, 16 hours, 
31 minutes, and not 6939| days, as already obtained. 

The error of l h 29® amounts to a day in 308 years. But the 
framers of the ecclesiastical calendar assumed this error to reach 
one day in 312| years, or 8 days in 2500 years. So they directed 
that a correction be made, such that M be diminished by 1 seven 
times successively at the ends of 300-year periods, and an eighth 
time at the end of a 400-year period, or 8 times in 2500 years. 
The last period of 2500 years terminated at the end of the year 
1799, and the correction was then 5 ; new corrections are therefore 
required in 2100, 2400, 2700, 3000, 3300, 3600, 3900, all at in- 
tervals of 300 years. But the next following correction does not 
come until 4300 instead of 4200, on account of the eighth period 
being one of 400 years. This condition will be satisfied for all 
time if we divide 8 c + 13 by 25, call the remainder r 8 , and subtract 
from M the correction : 

8 c + 13 - r 8 
25 

This may be verified readily by drawing up a table of this cor- 
rection, when it will be found to have the value 5 for y = 1799, 
and to increase thereafter forever in the proper way. We have, 
then, finally, for the Gregorian calendar : 

M = 15 + c - i (c - r0 - A (8 c + 13 - r 8 ) ; 
and wnen M is greater than 30, we may subtract from it, if we 
choose, the largest possible exact multiple of 30. And in the 
Gregorian calendar the date of Easter full-moon is now March 
21 -\- r 1} where r 7 is the remainder resulting from the division by 
30 of the number 19 r 6 + M. 

Having thus found a method of calculating the Gregorian date, 
March 21 + r?, of the Easter full-moon, we must now find the 
date of the Sunday next following, which will be Easter Sunday. 
We need therefore only calculate the week-day of the date March 
21 + r 7 , to know the date of Easter. Referring to our former 
civil calendar formulas, we shall find the remainder r 5 for the Easter 
full-moon date, March 21 + r 7 , which remainder we shall call 
rg for this special case, if we divide by 7 the quantity : 

5 r 3 + 3 r 4 + K + 2 + r 7 . 

33.) 



ASTRONOMY 

Now if r 9 comes out 0, the Easter full-moon comes on Sunday, 
and Easter is 7 days later, according to the Nicene decree. If r 9 is 
1, the full-moon day is Monday, and Easter is 6 days later. In 
general, we obtain the date of Easter Sunday by adding to 
March 21 -j- r 7 the number : 

7-r 9 . 

Collecting all our formulas, we can now find the date of Easter 
Sunday as follows ; and thus the rule given on p. 148 is demon- 
strated. 



Divide 


by 


and call 
remainder 




Century No., c 


4 


ri 


K = 5 n + r 2 , Gregorian calendar. 


Century No., c 


7 


V2 


K = 0, Julian calendar. 


Year No., y 


4 


n 


M = 15 + c - i (c - n) 
— is (8 c + 13 — r 8 ), Greg, calen- 


Year No., y 


7 


n 


dar. M = 15, Julian calendar. 


Sc + 13 


25 


n 


Easter Sunday is then March 


Year No., y 


19 


n 


28 + r 7 - r-9, 


19 n + M 


30 


r 7 


or April r 7 — r 9 — 3. 


+ 2 + r 7 j 


7 


r 9 


The following are values of K and 






M for the Gregorian calendar: 








1800-1899, K = 14, M = 23, 








1900-1999, K = 20, M = 24. 



As an example, let us calculate the date of Easter Sunday for 
1913. We have : 

n = 1, n = 2, K = 20, M = 24, r 6 = 13, r 7 = 1, r 9 = 6 ; 

Easter Sunday is on March 28 + 1 - 6 = March 23. 

We must now explain the two exceptions that occur in the 
Gregorian calendar only (p. 149). The first of these happens when 
r 7 = 29. The formulas have been deduced on the supposition 
that the March and April full-moons occur at an interval of 30 
days. But that interval may be 29 days only. The framers of 
the calendar have assumed, rather arbitrarily, that if there is a 
full-moon on March 19, or earlier in March, the April full-moon 

390 



APPENDIX 

will occur 30 days later. But if the March full-moon is on the 
20th, or later, the April full-moon will happen 29 days later. 
Thus the ecclesiastical April full-moon will happen on the same 
day, no matter whether the March full-moon comes on the 19th 
or 20th. 

As this cannot occur in reality, the framers of the calendar have 
directed that when the March full-moon happens on the 20th, 
which occurs whenever r 7 = 29, then r 7 shall be diminished arbi- 
trarily by 1. That is, we must use 28 instead of 29, or move the 
April moon back one day. But the diminution of r 7 by 1 will 
ordinarily also diminish r 9 by 1. Consequently, r 7 — r 9 will 
remain unchanged, and so will the date of Easter Sunday, which 
depends on r 7 — r 9 . Only when r 7 = 29 and r 9 = will the 
change of r 7 from 29 to 28 have any effect. For when r 9 = 0, 
a diminution by 1 will change it into 6, and r 7 — r 9 will be diminished 
by 7, making Easter exactly one week earlier. But when r 7 = 29 
and r 9 = 0, the rule always makes Easter come on April 26. 
Therefore the exception is as stated : whenever Easter comes by 
the rule on April 26, use April 19 instead. There will be an example 
of this in 1981. 

Unfortunately, the above exceptional case introduces another 
complication. The change of r 7 from 29 to 28 does not make it 
impossible for the value r 7 = 28 to occur again under the general 
rule, and during the same 19-year period. This might make two 
full-moons occur on the same date twice in a single 19-year 
period, which is, in fact, impossible. To avoid this, the framers of 
the calendar have ruled, again arbitrarily, that there shall be a 
second exception. Under this exception, 28 is changed to 27, 
whenever, in the same 19-year period, the first exception occurs. 

We must therefore investigate when the first exception can 
occur. In determining r 7 we performed a division by 30. Let us 
indicate the quotient of this division by v. Then we have, if r 7 
is 29, according to the first exception : 

19 r 6 + M = 30 v + 29. 

Now multiply this equation by 11, and add 11 to each member. 
This gives : 

209r 6 + 11M + 11 = 330*; + 319 + 11 = 330*; + 330. 

391 



ASTRONOMY 

The right hand member is now divisible exactly by 30 ; . there- 
fore the left-hand member is also so divisible. But the division of 
209 r 6 by 30 will leave a remainder of 29 r 6 . To make this dis- 
appear, the remainder in the division of 11 M + 11 by 30 must be 
r 6 . But r 6 is always less than 19 by its definition. Therefore the 
first exception will occur whenever, in the division of 11 M + 11 
by 30, the remainder is less than 19. 

But again, as in the case of the first exception, the change of 
r 7 from 28 to 27 will make no difference in the date of Easter, 
unless r 9 = 0. When r 7 = 28 and r 9 = 0, Easter, according to the 
rule, comes on April 25. The change of r 7 moves this date to 
April 18. Therefore the second exception reads : whenever, in 
the division of 11 M + 11 by 30, the remainder is less than 19, and 
if r 7 = 28 and r 9 = 0, Easter Sunday is on April 18, instead of 
April 25, as given by the rule. An example of this will occur in 
1954. 

To complete this subject it is necessary to remark that r 7 can 
never be 29, 28, and 27 within a single period of 19 years. There- 
fore no further exception is necessary on account of the possibility 
that the above two exceptions might, by acting together, produce 
two cases of r 7 = 27 in a single 19-year period. 

Note 17. The Sextant (p. 154). 

To prove the fundamental principles of the sextant, that the 
angle between the mirrors is half the altitude of the sun, imagine 
the plane of the paper to be the plane of the circle of the sextant. 
Then, in Fig. 113, the plane of the circle is supposed to be held 
vertically, in such a way that it will pass throngh the sun at S. 
The navigator sees the horizon with the upper part of the telescope 
through the unsilvered part of the mirror m ; and he sees the sun 
along the line TmMS after reflection from both mirrors. The 
angle MTm is the altitude of the sun above the horizon; the 
angle at P is the angle between the mirrors. It is necessary to 
prove that : 

Angle MTm = 2 angle P. 

The lines MP' and mP' are drawn perpendicular to the mirrors 
M and m. Then, according to the optical principles governing 

392 



APPENDIX 




Fig. 113. Theory of Sextant. 

the reflection of light from plane mirrors, the two angles a are 
equal and so are the two angles b. Furthermore, the angle SMm, 
or 2 a, is an exterior angle to the triangle mMT. Consequently : 

Angle 2 a = angle 2 6 + angle MTm, 
or : 

Angle MTm = 2 (angle a - angle b). 
Similarly, from the triangle mMP' : 

Angle P' = angle a — angle b. 

But angle P' = angle P, because their sides are perpendicular, 
each to each. Therefore, 

Angle P = angle a — angle b. 
And it follows that : 

Angle MTm = 2 angle P. 

Q. E. D. 

393 



ASTRONOMY 

Note 1 8. Longitude Determination (p. 158). 

A reference to Fig. 106 (p. 364) will show how the longitude may 
be computed from the sun's altitude, measured with the sextant. 
In the figure, if S' is the sun, the arc AS' is the measured altitude. 
If this be subtracted from 90°, we have the arc ZS' } or angular 
zenith distance of the sun. The arc S'D is the sun's declination, 
and may be ascertained for the date of the observation from the 
nautical almanac. Subtracting this declination from 90° makes 
known the arc PS', or the angular polar distance of the sun. 

The ship's latitude is also supposed to be known; without it, 
the longitude cannot be computed. But the ship's latitude 
always is known, because the navigator will have determined it 
at noon, and can easily allow for any slight change in the ship's 
latitude since the last noon observation, since he knows the com- 
pass course he is steering, and the approximate speed of the ship. 

But the latitude is the arc PN in the figure, or the altitude of 
the celestial pole above the horizon. This latitude being subtracted 
from 90°, gives the arc ZP, or the angular distance from the ce- 
lestial pole to the zenith. Thus these three subtractions from 90° 
make known the three sides of the spherical triangle ZPS'. 

It is a principle of trigonometry that any spherical triangle can 
be solved completely, and all its parts made known, if we know 
its three sides. Thus we find the spherical angle S'PZ, of which 
the vertex is at the pole, and which is measured by the arc DM on 
the celestial equator. But DM is by definition the hour-angle of 
the sun S' ; and the sun's hour-angle is the local apparent solar 
time. This need merely be corrected by applying the equation of 
time (p. 134) to obtain the local mean solar time of the ship, ready 
for comparison with the Greenwich time taken from the face of 
the chronometer by an assistant at the instant when the sun was 
observed for longitude by the navigator. 

Note 19. Moon's Distance (p. 169). 

Figure 114 shows how the moon's distance is determined. We 
shall assume, as a sufficiently close first approximation, and to make 
the problem easy to understand, that the two observatories are 
situated on the same meridian of terrestrial longitude, but very 
widely separated in latitude. One should be in the northern 

394 



APPENDIX 




hemisphere; the other in the southern. The observatories of 
Greenwich, England, and the Cape of Good Hope, for instance, 
satisfy these conditions quite closely. 

In Fig. 114, then, and 0' are the two observatories, the circle 
representing the earth. The arc 00' is known, for it is simply 
the latitude difference of the 
two observatories. The angle 
OCO f is equal to the arc 00' ; 
and [the lines CO and CO' are 
each known radii of the earth. 
Therefore, by simple trigo- 
nometry, we can solve the tri- 
angle OCO', and gain a knowl- 
edge of the distance 00' , which 
is to be our base-line, and of 
the two angles COO' and CO'O. 

We next measure at both 
observatories simultaneously, 
with suitable astronomic in- 
struments, the exact lunar alti- \^ 
tude, or angular elevation of ^\z' 
the moon above the horizon, at FlQ 114 Moon , s Distance . 
the instant when the diurnal 

rotation has brought the moon to the celestial meridian. These 
simultaneous observations will be possible, because the moon will 
reach the meridian of both places at the same instant, since we 
have imagined our two observatories lying on the same meridian 
of terrestrial longitude, and therefore having the same celestial 
meridian over them in the sky. 

Having measured the moon's altitude above the horizon, we can 
at once find its angular distance from the zenith. For the latter 
point is always 90° distant from the horizon; so that we obtain 
the angular zenith distance of the moon by simply subtracting 
its measured altitude from 90°. 

Those two angular zenith distances, thus known from the 
measured altitudes, are the angles MOZ and MO'Z' in Fig. 114. 
Next we subtract these angles from 180°, giving us the angles MOC 

395 



ASTRONOMY 

and MO'C. From these we again subtract the angles COO' and 
CO'O, found above, thus obtaining values of MOO' and MO'O. 
These now make possible a trigonometric solution of the triangle 
MOO', of which we now know the base 00' and the two adjoining 
angles. Thus we get OM and O'M in miles. After that we can 
solve the two triangles COM and CO'M, since we know the length 
of the two sides CO and OM, as well as the included angle COM ; 
and in the other triangle we know CO' and O'M as well as the in- 
cluded angle CO'M. A solution of either triangle gives us CM, 
the distance from the center of the earth to the moon. 

It is scarcely necessary to add that the ideal condition here 
assumed as to location of observatories does not exist in fact. But 
a slight divergence from this condition in no way impairs the prin- 
ciple of the method; it merely adds a certain additional com- 
plexity to the trigonometrical calculations. 

Note 20. Lunar Parallax (p. 169). 

Figure 42 shows that the moon's parallax and distance are con- 
nected by a very simple trigonometric formula : 

sin parallax =4g= radius of earth 



CM distance of moon 

This formula shows that we can calculate the parallax if we know 
the distance, or the distance if we know the parallax. The two 
are closely related; astronomers frequently speak of measuring 
the parallax of the moon or other heavenly body, when they 
merely mean a measurement of its distance. 

Note 2i. The Moon's Mass (p. 175). 

Figure 115 is intended to make this matter clear. S is the sun ; 
the circle is the annual terrestrial orbit. When the center of 
gravity is at d, the earth at E±, and the moon at Mi, the sun will 
appear from the earth projected in the direction &. This is exactly 
the same as would be the case if there were no moon, for then the 
earth would itself be at &. But when the center of gravity is at 
C 2 , the earth will be at E 2 ; and the sun will be seen projected in 
the direction S 2 ', instead of S 2 , which is its direction as seen from 
C 2 , and which would be its direction from the earth if there were 
no moon. 

396 



APPENDIX 



Thus the sun will be seen a certain angular distance in advance 
of its proper position ; and a half-month later it will be similarly 
retarded. The total range is 12", so that the angle S2SS2' is 6". 
Therefore, in the triangle C 2 SE 2 , we know the angle C2SE2 to be 
6" ; and we know 



C 2 S 
dis- 
the 




Fig. 115. Mass of the Moon. 



the two sides 
and E 2 S, the 
tance from 
earth to the sun, 
which can be meas- 
ured. Solving 
the triangle, we 
find the side C 2 E 2 
to be about 2880 
miles. We then 
form the propor- 
tion : 

E2C2 : M 2 C 2 = 
moon's mass : 
Earth's mass ; 

from which we can 
compute the lunar mass, since the other quantities in the propor- 
tion are now all known. 

Note 22. Concavity of Moon's True Orbit with Respect to the 

Sun (p. 181). 

We can test this question by means of Fig. 116. It is evident 
from a mere glance at Fig. 46 (p. 181) that there is no doubt as to 
the concavity of the moon's orbit toward the sun at the time of 
full-moon, shown at M 3 . Difficulty arises only in the case of the 
new-moon phase, shown at Mi and M b . Therefore, in Fig. 116, 
we shall examine especially the new-moon phase. Let E h Mi, and 
S be positions of the earth, moon, and sun at the time of new-moon. 
Let EiE 2 be a portion of the earth's orbit around the sun ; and let 
the small circles represent the lunar orbit around the earth. While 
the earth moves from Ei to E 2 , we may suppose the moon to move 
around the earth from C to M 2 . In other words, if the moon did 

397 



ASTRONOMY 



not revolve around the earth, it would be at C when the earth 
reached E 2 . Designate the angle EiSE 2 by the letter 0, and let 
r e and r m represent radii of the earth's orbit around the sun and 
the moon's orbit around the earth. Finally, let E X T be a tangent 

to the earth's orbit at 
/Y~^\ E\) draw E 2 P perpen- 

-t dicular to EiS; E 2 B 
parallel to EiS; and 
M 2 A parallel to EJ 1 . 
If we let be a small 
angle, MiM 2 will be a 
small part of the moon's 
path near new-moon : 
it will evidently be con- 
cave towards the sun if 
M 2 is farther from the 
tangent EiT than is Mi. 
While the moon was 
moving from Mi to M 2 
the entire lunar orbit 
fell away from the tan- 
gent the distance EiP; 
but the moon rose toward the tangent a distance nearly equal to 
AB. Therefore the moon recedes from the tangent a total dis- 
tance of EiP — AB. Now we have, evidently : 




Fig. 116. Moon's True Orbit. 



But: 
Also 



EiP = r e - r e cos 0, 
AB = r m — r m cos MiE^A. 
M 2 E 2 A = M 2 E 2 C + CE 2 A. 



(1) 
(2) 



M 2 E 2 C = 13 0, because the moon's angular motion in its orbit 
is about 13 times as fast as the earth's (p. 161) ; and : 

CE 2 A = 6, because AB is parallel to EiS. 



It follows that: 

and, from equation (2) : 



M 2 E 2 A = 14 0; 

AB = r m — r m cos 14 0. 
398 



(3) 



PHILOSOPHIC 

NATUR-ALIS 

PRINCIPI A 

MATHEMATICA 



Autore J S. NEWTON, Inn. Coll. Cantab. Sac. Mathefeos 
Profeffore Lucafiano y <k Societatis Regalis Sodali. 



IMPRIMATUR- 
S' P E P Y S, Reg. Soc. P R £ S E & 



L N D I N J ? 

Juflii Societatis Kegia ac Typis Jofephi Streater. Proftat apud 
plures Bibliopolas. Anno MDCLXXXVII. 



PLATE 32. Title-page of Newton's Principia. 



APPENDIX 



A simple calculation, using the value 6 = 1°, r m = 240,000, 
r 2 = 93,000,000, gives : 

EJ> = 16,000 miles, 
AB = 7130 miles. 

It follows that the moon recedes from the tangent about 8870 
miles in one day, while the earth is moving about 1° in its orbit 
around the sun. This proves that the moon's true orbit is con- 
cave towards the sun, even at the time of new-moon. 

Note 23. Law of Areas (p. 186). 

Figure 117 shows how the point P 3 ' is found. Draw P3P3' 
parallel and equal to P2P2 '. Then the actual motion of the planet 
in the second second will take place along the 
diagonal P2P3' of the parallelogram P2P3P3P2. 
This theorem of the " parallelogram of forces" is 
demonstrated in works on elementary physics : l 
perhaps the easiest way to understand it is to 
notice that P 3 ' is point to which P 2 must go, if 
it actually completes separately the two motions 
P 2 P 2 ', and P 2 / P3 / equal and parallel to P2P3. 




117. Law of 

Areas. 



Note 24. Law of Areas (p. 186). 2 

We have still to prove the triangles &P1P2 
and SP2P3 equal in area. Referring again to 
Fig. 117, we see that the triangles SP2P3 and SP2P3 are equal, since 
they have the same base SP 2 , and their altitudes are equal because 
their vertices P 3 and P3' lie on the line P3P3, which is parallel 
to P 2 $. And we have already found the triangle SP2P3 equal to 
SP1P2. Therefore the triangle &P2P3' is also equal to &PiP 2 . 

1 Figure 117 may be found in the first edition of Newton's immortal Prin- 
ciple/,, of which the title-page is reproduced as Plate 32. The president of 
the Royal Society, whose name appears on the title-page as having author- 
ized the printing, is the famous diarist. On p. 13 of the Principia ap- 
pears Corol. I : " Corpus viribus conjunctis diagonalem parallelogrammi 
eodem tempore describere, quo latera separatis." 

2 On p. 37 of the same work of Newton appears Prop. I, Theorem I : 
"Areas quas corpora in gyros acta radiis ad immobile contrum virium 
ductis describunt . . . esse temporibus proportionates. " 

399 



ASTRONOMY 



Note 25. Harmonic Law (p. 188). 

It would carry us too far afield in mathematical astronomy to 
give here the demonstrations by which Kepler's three laws may be 
derived from Newton's single law ; but there is little difficulty in 
considering by elementary methods the special case of a circular 
planetary orbit. The circle is, in fact, a close approximation to 
the actual planetary orbits in the solar system : none of these orbits 

are very much flattened from the 
circular form. 

We must first investigate the 
nature of the solar attractive force. 
In the case of a circular orbit this 
force is necessarily constant under 
Newton's law, because the planet 
is always at the same distance 
from the sun. Now consider the 
accompanying Fig. 118. Let PP' 
be a very short arc of a circle, 
whose center is at S. Draw the 
diameter PD and the chord P'D; 
and let fall the perpendicular P'C 
upon PD. Draw the chord PP' , the tangent PP" ; and let fall 
the perpendicular P'P" upon PP" from P '. Then, from the 
similar right-angled triangles PP'C and PP'D, we have : 




Fig. 1 18. Solar Attraction. 



PP' : PC = PD : PP' 



or: 



PC = 



PP' 
PD 



Now let our circle be a planetary orbit, with the planet at P, 
the sun at S ; and suppose that in one second of time the planet 
would move along the orbit to P'. We may consider this very 
short arc PP' coincident with its chord PP' . 

From the principle of the parallelogram of forces (p. 399), the 
actual motion PP' may be regarded as the resultant of two 
motions : PP" , which would be the planet's actual motion from 
P in a second if the solar attraction were to cease suddenly ; and 

400 



APPENDIX 

PC, which would be the planet's actual motion in a second if 
attraction toward the sun operated alone. 

Now PP' is the planet's velocity in its orbit per second, which we 
shall call V ; and PD is twice the radius of the orbit, which latter we 
shall call r. Let us also designate the distance PC by the letter x, 
and consider all distances to be measured in miles. Then, from 
the geometry of the figure, as we have just seen : 

•_-£• a) 

But as we have said, PC or x is the distance the planet would 
move or "fall" toward the sun in a second, if the solar attraction 
acted alone, without any additional orbital or tangential impulse 
derived, perhaps, from the original catastrophe by which the 
planet was brought into existence. The question now is : How 
great must be the solar attractive force to cause a planet to fall 
from a position of rest at P through the distance PC or x in 
a second ? 

This raises the question of how forces are measured. What is a 
suitable unit of force? Now the solar attraction is not applied 
suddenly as a single impulse ; it is applied continuously. Conse- 
quently, the planet would fall the short distance x toward the sun 
with a uniformly increasing velocity, faster and faster, but begin- 
ning with zero velocity at P. Its average velocity would be 
attained halfway between P and C. But the actual distance it 
would move in a second is of course the same as if it traveled 
constantly with its average velocity. And as it would fall a 
distance x miles in a second, its average velocity must be x miles 
per second. Therefore it would be moving with the velocity 
x miles per second when halfway between P and C; and upon 
reaching C its velocity would have increased to 2 x miles per second. 
But in astronomy, as in mechanics, our units are so chosen that 
force is always measured by the quantity of velocity accumulated 
in a second, multiplied by the quantity of mass in the moving body. 
The velocity thus accumulated in a second is called "acceleration" ; 
and as the velocity of the falling planet increased from zero to 2 x 
miles per second, the acceleration produced by the solar attractive 
2d 401 



ASTRONOMY 

force must be represented by the number 2x. Calling this ac- 
celeration /, we thus have : 

f=2x; 

and this, combined with equation (1), gives : 

/■~r (2) 

Now the whole circumference of the circular orbit is 2 irr ; and 
the planetary orbital velocity V is of course equal to the circum- 
ference divided by the period of orbital revolution. It follows 
that if we call this period t, expressed in seconds of time, we shall 
have: 

v ='*!L. 
t ' 

and, therefore, from equation (2) : 

/= *A (3) 

If we now apply equation (3) to two separate planets, indicating 
by subscript numbers quantities belonging to the first and second 
of the two planets, we shall have : 

or: 

h r 2 k 2 

But we know from Newton's law that the attractive forces 
exerted by the sun on two different planets, if of equal mass, will 
be inversely proportional to the squares of the distances separating 
those planets from the sun. This may be written thus : 

/i:/ 2 = r 2 2 :r 1 2 ; 
or: 

£ = ^ 2 . (5) 

A n 2 v J 

Equating the right-hand members of equations (4) and (5) gives : 

if : t 2 2 = n 3 : r 2 3 . 

402 



APPENDIX 

This is the third (or harmonic) law of Kepler, which is therefore 
thus demonstrated as a consequence of Newton's law in the case 
of circular orbits. A similar proof is possible, by the aid of the 
higher mathematics, without this assumption as to the form of 
the orbit; but a small correction is always required, because we 
have taken the planets to have equal masses. 

Still retaining our circular orbit formulas as a sufficient first 
approximation, we are now in a position to understand Newton's 
famous test as to whether the force of gravitation observable on 
the earth also extends outward as far as the moon. We shall 
present this test here in a somewhat modernized form, based on 
the formulas just obtained. Resuming our equation (3), we have 
the acceleration exerted by the earth upon the moon : 

/=4* 2 £, (6) 

in which r is now the distance from the earth to the moon, and t the 
moon's sidereal period (p. 161). This equation is correct, if the 
Newtonian law of gravitation extends to the moon, and not other- 
wise. 

Newton's test now consists in comparing the value of / calculated 
by means of equation (6) with its value easily obtained by another 
method. It was known from laboratory experiments, even in 
the time of Newton, that the earth attracts an object situated on 
its surface with a force which is called the " force of gravity," and 
which produces an acceleration designated by the letter g in physics. 
It is also known that the earth's attraction upon any object 
exterior to it acts as if the entire mass of the earth were concen- 
trated at its center. 1 

Now the distance from the earth's center to an object on its 
surface is equal to the earth's radius, and may be designated by 
R; while the distance from the earth's center to the moon is r. 
It follows that if the earth's attraction varies inversely as the 
square of its distance from the object attracted, as postulated by 
Newton, we may write the following simple proportion involving 

1 This was demonstrated by Newton. 
403 



ASTRONOMY 

/, due to the earth's attraction upon the moon, and g ; due to the 
earth's attraction upon surface objects : 



j y r 2 R2 > 



from which we have at once 



/=JT CO 

r 2 

The values of / in equations (6) and (7) must be equal, if both g 
and /result from the same identical law of Newtonian gravitation. 
Equating these quantities gives : 

«^=4^; 

y r 2 t 2 ' 

or: 

a = 47r 2 ^-. (8) 

y t2R2 w 

In this equation, r is the moon's distance from the earth, which 
we here suppose expressed in feet; and t is the moon's sidereal 
period, in seconds of time. Let us then calculate g, and ascertain 
whether it agrees with its known value derived by physicists from 
laboratory experiments. The moon's sidereal period is 27 d 7 h 43 m 
11.5 s , or 2360591.5 seconds. The moon's distance, r, is 238,840 
miles, or 1,261,075,200 feet. The earth's radius is 3858.8 miles, 
or 20,902,464 feet. The value of v is 3.1416. Making the calcu- 
lation by means of logarithms, the above data give, by the aid 
of equation (8) : 

9 = 32.5, 

which is in very close accord with the value of g found directly by 
experiment in the physical laboratory. It is a most astounding 
thing that a series of quantities can thus be brought together, 
as it were, from various parts of the solar system : the moon's 
distance determined by astronomic observations at Greenwich 
and the Cape of Good Hope (p. 395) ; the earth's radius by triangu- 
lation measures (p. 97) ; the moon's period by noting the interval 
between distant full-moons (p. 162), — it is astounding that these 
heterogeneous quantities, thus determined by direct observation, 
can be combined by a simple formula based on that wonderful law 

404 



APPENDIX 

of Newton, and made to produce the identical value for g which 
we obtain by terrestrial laboratory observations, quite without 
using astronomic material. There could not be a more striking 
proof of the unity of science ; nor can any doubt remain that the 
same force of gravity which controls experiments on the earth, 
also controls the moon's orbital motion. 

Note 26. Planet's Mass (p. 205). 

Let us suppose once more that orbits are all circular. Consider- 
ing the satellite orbit, we found, when discussing Newton's test 
of the law of gravitation by means of the moon, that the accelera- 
tion due to the attractive force toward the center of the orbit may 
be represented by the equation (p. 403) : 

where r is now the radius of the satellite's orbit in miles, and t its 
period of revolution. In this equation, / is due to a continuously 
acting attractive force toward the planet situated in the center 
of the orbit, supposed circular. 

It is easy to obtain another expression for this force. We have 
at once, from Newton's law of gravitation (p. 376), that the attrac- 
tion existing between the planet and the satellite is proportional 
to the product of their masses, and inversely proportional to the 
square of the orbital radius. If we let M indicate the planet's mass, 
and m that of the satellite, this force is : 

n Mm 
r 2 

where G is a constant depending on the units adopted for linear 
distances, etc. 

Now this Newtonian force produces the acceleration / in the 
planetary mass m ; and since force is measured by the acceleration 
produced, multiplied by the mass moved, it follows that : 



Mm 



fm = G 

r 

or: 

r 2 
405 



ASTRONOMY 

and this is the acceleration due to the attraction of the planet on 
the satellite. 

In an exactly similar way, we can show that the satellite pro- 
duces an acceleration of the planet equal to : 

r 2 
so that the total acceleration existing between the two bodies is : 

If we now equate this value of the acceleration to that given 
in the equation for /, we have : 

G K±jn = ^L 
r 2 t 2 

or : 

Let us next apply this equation to two planets, each having a 
satellite, and indicate by subscript numbers quantities belonging 
to the two planets. We thus easily obtain the proportion : 

rf 

Mz + rrh r^ 

or: 

M 1 + m 1 :ilf 2 + m 2 =^:^. 
h 2 k 2 

With the help of this general proportion, we can now find the 
planet's mass as compared with that of the earth. . We need only 
let the subscript 1 refer to the earth and moon, and the subscript 2 
to the planet and satellite. Then everything is known in the 
proportion except M 2 + m 2 , if we have determined by direct 
observation the distance and period of the satellite with respect to 
its planet. It is to be noted that this method gives only the sum 
of the masses of the planet and its satellite, not the mass of the 
planet alone. But this is of minor importance, since the satellites 

406 



APPENDIX 

are almost always very small compared with their planets : and, 
in any case, it is the combined mass of the system, including both 
planet and satellite, that we really need to know. For it is this 
combined mass which pulls upon other bodies in space ; and it is 
the pulling force upon such other bodies which must be used in 
any further calculations relating to orbits, etc. 

When a planet has no satellite, as in the case of Venus and Mer- 
cury, we cannot employ the above simple and accurate method. 
We must then have recourse to a mathematical discussion of the 
slight perturbations the planet produces in the observed motions 
of other bodies in the solar system. These perturbations, of course, 
depend on the planet's mass, being greater for a massive planet 
than for a small one ; and consequently the planetary masses must 
admit of numerical evaluation from the observed perturbative 
effects they produce. Unfortunately our knowledge of the mass 
of Mercury is still incomplete ; that of Venus, however, is known 
with some precision. 

The mass of a planet once determined, it is easy to calculate the 
force of gravity on the planet's surface, its Superficial Gravity, 
as it is called. If we designate by P the planet's radius in terms 
of the earth's radius, and by g the planetary superficial gravity, 
analogous to the customary designation of the force of gravity on 
our earth's surface, we have at once, from Newton's law of gravi- 
tation : 

M 

where M is the planet's mass in terms of the earth's mass. 

To ascertain the planet's density in comparison with that of 
our earth, we proceed thus : We know, in general : 
Mass = Volume X Density. 
Therefore we have for the earth's mass M, : 

M. = FA, 
where A e represents the terrestrial density, and V, the earth's 
volume. 
And for the planet we have : 

1V± p V pl-L p . 

407 



ASTRONOMY 

Consequently : 

W e M.' 
But, again using P to indicate the planet's radius : 

Therefore, if we take the mass of the earth as unity : 

A = Me A 

If we wish the actual specific gravity of the planet, compared 
with water, we must substitute for the A e the value 5.53, as de- 
termined by means of the Cavendish experiment (p. 110). 

Note 27. Synodic and Sidereal Periods (p. 209). 

Let us indicate by J m and 2jJ sld the sidereal periods of Jupiter 
and the earth, each expressed in mean solar days. E ala , for in- 
stance, is then 365i, approximately. Then, regarding both orbits 
as circular, and the motions uniform, the earth in one day will pass 

360° 
over a fraction of its total orbit represented by — — ; and Jupiter 

360° 
will pass over a fraction represented by —— • These two fractions 

•Add 

are not equal : if we take the difference : 

360° 360° 

E s \d J sid 

this quantity will be the angle by which the earth and Jupiter 

fail to lie in a straight line, as seen from the sun at the end of one 

day after the beginning of Jupiter's synodic year (see Fig. 55, 

p. 208). 

This quantity is therefore, by definition, Jupiter's daily synodic 

motion. But if Jupiter's entire synodic period be represented 

360° 
by Jsynj its daily synodic motion will also be — — Equating this 

*i syn 

with the above value of the same quantity, we have : 
360° 360° 360° . 



r syn -Esid 

408 



APPENDIX 

or: _1_ = J 1_. 

t/syn -&sid «^sid 

By means of this equation, Jupiter's synodic period may be 
calculated from his sidereal period, and vice versa; for E BiA is 
known to be 365J days. 

Note 28. Periods of Inferior Planet (p. 210). 

The synodic motion, as in the case of a superior planet, again 

depends on the earth's orbital motion as well as on that of the 

planet. As before, the daily sidereal motions of Venus and the 

360° 360° 

earth may be represented by — — and — — The difference will be 

V aid -Esid 

the daily synodic motion of Venus, supposed seen from the sun. 
This quantity is : 

360° 360° 

^sid -E'sid 

Thus the formula for the daily synodic motion of an inferior 

planet is perfectly analogous to that for a superior planet, except 

that the terms are now interchanged. This is of course due to 

the fact that the superior planet has a slower angular motion 

around the sun than the earth, while the inferior planet has a 

faster angular motion. But, as before, if F ayn be the synodic 

360° 
period of Venus, the daily synodic motion will be — — ; and we 

'syn 

have : 



or : 



It follows that for any planet whatever the reciprocal of the 
synodic period is always equal to the difference between the 
reciprocals of the planet's sidereal period and the earth's sidereal 
period of 365J days. 

Note 29. Table of Periods (p. 211). 

It will be of interest to calculate some of the numbers in the 
table (p. 211) by means of the formulas in Notes 27 and 28. We 
find: 

409 



360° 


360 p 


360° 


v syn 


v sid 


E s m 


1 

^syn 


1 

V s[d ~ 


1 

-Esid 



ASTRONOMY 





Reciprocal 

of Sidereal 

Period 


Reciprocal 

of Earth's 

Sidereal 

Period 


Difference 


Synodic 
Period 


Mercury 

Mars 

Uranus 


.011364 
.001456 
.000033 


.002738 
.002738 
.002738 


.008626 
.001282 
.002705 


116 
780 
370 



In computing the numbers in this table, all periods have been 
reduced to days ; and the numbers in the last column are recipro- 
cals of those in the column headed " Difference." It is at once clear 
from this little calculation how the peculiarities of the table of 
periods arise. As the sidereal periods of the outer planets increase, 
the reciprocals of these periods must diminish. Consequently, 
these reciprocals must gradually approach zero, and the numbers in 
the column " Difference " must approach the value .002738. So the 
numbers in the final column of synodic periods must approach the 
value 365i, or the earth's period. This is just what we should 
expect. For the outermost planets remain practically stationary 
for many days among the fixed stars, and must therefore have a 
conjunction every time the earth goes around its orbit, or very 
nearly so. The effect of their own slow orbital motion on their 
synodic motion is necessarily very slight. 

Note 30. Greatest Elongation, Mercury and Venus (p. 212). 

In Fig. 58 (p. 212) the triangle SEV is right-angled at V. We 
can therefore calculate the angle SEV, which is the required greatest 
elongation angle, by means of the formula : 



■mSBF-Jg, 



or: 



sin of greatest elongation 



distance of planet from sun 
distance of earth from sun 



Let us make the calculation for Mercury. The orbit of this 
planet is more flattened than any other in the solar system : 
the approximate distance of Mercury from the sun varies from 28.5 

410 



APPENDIX 



to 43.5 million miles. Obviously, the greatest elongation will be 
larger if it happens when the planet is in that part of its orbit which 
is farthest from the sun. We shall therefore make the calculation 
twice, using the two values just given for the distance from Mercury 
to the sun. We have : 



Distance of Mercury . . 
Distance of earth . . . 
Log distance of Mercury . 
Log distance of earth . . 
Log sin greatest elongation 
Greatest elongation . . 



Least 



28.5 

93.0 
1.4548 
1.9685 
9.4863 
17°51' 



Greatest 



43.5 

93.0 
1.6385 
1.9685 
9.6700 
27°53' 



From this calculation we see that Mercury can never attain 
an angular distance from the sun greater than 28°, as seen projected 
on the sky from the earth ; and ordinarily its greatest elongation 
will be much less than 28°. 

Note 31. Temperature of Mars (p. 226). 

The distance from Mars to the sun is about lj times that from 
the earth to the sun. Therefore, if we assume the heat radiated 
by the sun to diminish with the square of the distance, Mars 
1 



receives only 



as much heat as the earth, or f as much. We 



(1.5)2 

may also assume that, on the average, all planets radiate annually 
the same amount of heat they receive ; otherwise they would be- 
come continuously hotter or colder. Now we have a law of physics 
known as Stefan's law, which gives us an estimate of the quantity 
of heat a body will radiate at different temperatures. According 
to this law, calling the quantity of radiated heat Q, and the tem- 
perature F (Fahrenheit), we have : 

for the earth, Q e = (458° + F e )\ 
for Mars, Q u = (458° + F n )\ 
But if each planet radiates as much heat as it receives, 

ft. = 2. 

411 



ASTRONOMY 

Therefore : (458° + F e Y = 9 

(458° + i^) 4 4* 
Now for the average temperature of the earth, we may put 

F e = 60°. 
Therefore : 

_P^_ = |, (458° + F m y = U51S Q y, 

458° + F m = ^1(518°) = 0.82 X 518° = 425°. 
So that : 

F m = - 33° Fahrenheit. 

This result is of course uncertain, because we cannot be sure 
that Stefan's law is really reliable in the case of Mars and the 
earth. It has been tested in the laboratory only, and for a black 
body radiating its heat freely. 

Note 32. Saturn's Ring (p. 245). 

We have already found (p. 402) a formula for the accelera- 
tion toward the center of an orbit. It is : 

f = Yl. 

r 

But, according to Newton's law, / is inversely proportional 
to r 2 ; so that V 2 must be inversely proportional to r. Therefore 
if the rings are really a mass of satellites, the squares of their 
linear velocities are inversely proportional to their distances from 
the planet. In other words, the outside of the ring should revolve 
more slowly than the inside. 

The outside radius of the ring has been measured by the usual 
methods (p. 203) to be 86,500 miles ; the inner, 55,700. The square 
roots of these numbers are in the ratio of 1 to 1.24 ; while the ob- 
served linear velocities are in the ratio of 1 to 1.25. There is 
therefore a surprisingly close agreement ; and there can be no doubt 
that the various parts of the rings rotate in accordance with 
Kepler's harmonic law, and are composed of satellite swarms. 

Note 33. Halley's Transit of Venus Method (p. 269). 

We must first show how to calculate the length of the chord in 
seconds of arc. In Fig. 119, let 'S, Vi, and E be the positions of the 

412 



APPENDIX 

sun, Venus, and the earth at the moment of inferior conjunc- 
tion. Let P be the synodic period (p. 208) of Venus, in days. 
Then Venus gains a whole revolution of 360° on the earth 
in P days, from the defini- 
tion of the synodic period. 
In one day Venus gains 

— — • Therefore, if we let s" 

Fig. 119. Halley's Method. 

the arc ViV 2 represent the 

synodic gain of Venus on the earth in a day, as seen from the 

sun, we have: 




Angle S 



360 



But in the plane triangle SEV 2 , we have : 

sin S:smE = V 2 E:V 2 S, 

since the sines of the angles of any plane triangle are proportional 
to the opposite sides. 

Therefore: sinE= M sinS . 

V 2 E 

V s 

But the ratio -z-^— is known from the known relative lengths of 
V 2 E 

the radii of the two orbits belonging to Venus and the earth (cf. p. 
262). The angle S being also known, as has just been shown, it 
follows that we can calculate the angle E, which is the angular 
distance through which Venus advances across the face of the sun 
in one day, as seen from the earth. This angle is transformed into 
seconds of arc ; and the observers having found the fraction of a 
day required by Venus to traverse the observed chords, we find 
at once by proportion the lengths of the chords in seconds of arc. 
As soon as the lengths of the two chords SP and sp (Fig. 73, 
p. 269) thus become known in seconds of arc, the further proceedings 
are simple. For the angular semi-diameter, or radius, of ;the sun's 
disk is of course known also in seconds of arc (p. 118); conse- 
quently, it is possible to calculate the distances Sa and Sb (Fig. 73) 
in seconds of arc, and also their difference ab. We also know 
(Fig. 73) the ratio of the lines VA and Va, because we know 

413 



ASTRONOMY 



the relative distances of Venus and the earth from the sun. Va is 
0.723 if Aa is 1.000. Therefore : 

Va : VA = 723 : 277 ; 

and ab, in miles, is fff A B, provided, of course, that the distance 
AB is perpendicular to the plane of Venus' orbit. If not, it is 
easy to calculate the necessary correction. Now, knowing 

ab, on the sun, both in 
miles and in seconds 
of arc as seen from 
the earth, we easily 
obtain the distance of 
the sun. The simple 
Fig. 120 shows how this is done. Calling r the radius of the earth's 
orbit, or the distance from the earth to the sun, we have, from the 
right-angled triangle Eab, in which the line ab is on the sun, as 
usual : 

tan a6(seconds of arc) = ^(™ks) 

r (miles) 




Fig. 120. Halley's Method. 



or 



rftnitaO- ' a y M ^ 
tan ab (seconds) 



Note 34. Solar Parallax from the Aberration of Light (p. 271). 

Let us study somewhat in detail 
the action of light aberration. In 
Fig. 121, suppose that an observer at 
t has his telescope pointed in the 
direction tT ; that the earth, carry- 
ing the observer and telescope, is for 
the moment moving in its annual 
orbit in the direction tt', with the 
velocity v miles per second. Now 
suppose light from a star at S reaches 
T at the moment when the telescope 
is in the position tT. And suppose 
this light travels with a velocity of V 
miles per second in the direction ST. 

Now indicate by a the angle tTt'. Then we may say, as it were, 

414 




Fig. 121. Solar Parallax from 
Aberration of Light. 



APPENDIX 

that if the velocities v and V are properly proportioned to fit the 
angle a, the light will "stay in the telescope tube" while the tube 
is moving from tT to t'T', We shall then have : 

tan a = — • 

This equation signifies that a star at S will really appear pro- 
jected on the sky in the direction t'T'. In other words, the aberra- 
tion of light displaces the apparent position of the star on the sky 
through the angle a. 

And there is no difficulty in measuring this angle a: for the 
displacement of the star is always in the direction of the earth's 
motion, here W. And as that motion takes place in a nearly 
circular orbit, the displacement a must be in opposite directions 
at intervals of half a year (cf. p. 137). For the earth's orbital 
motion is, of course, reversed in direction at opposite points of 
the orbit. We have therefore merely to determine by observation 
the apparent declination of a star on the sky at intervals of six 
months. If a suitably located star is selected, the declination will 
be found to vary by twice the angle a; about 41" of aro. 

From this we easily compute the solar distance. For the 
velocity of light, V, is known from laboratory experiments. With 
V and a both known, we can compute v with the equation just ob- 
tained, and v is the earth's linear velocity in its orbit. Thus it 
has been found that v is about 18.5 miles per second. This we have 
now to multiply by the number of seconds in a year, to get the 
linear circumference of the earth's orbit. Finally, dividing by 
2 ir, we have the orbital radius, or the solar distance. 

Note 35. Sun's Mass (p. 291). 

To ascertain this quantity, we resume the formula which ex- 
presses the acceleration which the sun gives a planet. It is 
(p. 402): 

f =Y1 — (velocity of planet in orbit) 2 1 

r radius of planetary orbit 

In the case of the earth r is 93,000,000 miles. Assuming the orbit 

approximately circular, we can find its circumference by the formula 

Circumference = 2 irr ; 

415 



ASTRONOMY 

and this being divided by the number of seconds in a sidereal 
year, we find V, the linear orbital velocity of the earth, in miles 
per second. It is approximately 18|. Now, calculating /, we find : 

/ = 0.233 inch. 

If we now let g, as usual, represent the constant of terrestrial grav- 
ity, we may write a simple proportion by the aid of Newton's law : 

- _ sun's mass . earth's mass 
(sun's distance) 2 ' (earth's radius) 2 

This proportion is a direct consequence of Newton's law, which 
makes attractive forces proportional to masses, and inversely 
proportional to squares of distances. The earth's radius becomes 
the distance for terrestrial gravity g, because the earth attracts as 
if its mass were concentrated at its center; and the radius is 
the distance from the center to the surface, where gravity acts. 

In the proportion everything is known but the solar mass : we 
can therefore readily calculate it. 

Note 36. Angle at Earth's Center for Possible Eclipse (p. 300). 

To find the size of the angle MicS in Fig. 84, we consider the tri- 
angle O'MiC, taking the point M\ as the point of tangency of the 
moon at M 1 with the line O'O. Then, in the triangle O'MiC : 
sin M lC Q' = MiO 7 
sin MiO'c Mic 
But, as the sines of these small angles are proportional to the angles 
themselves, we may write : 

M idT = MiO' 
MxO'c Mic ' 
But MiO' = O'O - MiO = 93,000,000 - 240,000, very nearly ; 

Mic = 240,000. 
MicO' = 93000000 - 240000 = 3g6 
MiO'c 240000 

But MxO'c = solar parallax = 8".8. 

M x cO' = 8".8 X 386 = 57 r ; 
also O'cS = sun's angular radius as seen from the earth 

= 16', approximately. 
416 



APPENDIX 

Therefore : 

MicS =MicO' + 0'cS = 57' + 16'= li°, approximately. 

And if we now consider Mi to be at the center of the moon, the angle 
MicS will be increased by the moon's angular radius as seen from 
the earth, or 16'. So that, finally, the angle at c between the 
centers of the sun and the moon at Mi is lj° + 16', or 1J°, approxi- 
mately. 

Note 37. Draconitic Period (p. 305). 

We have seen (p. 299) that the moon's node makes a complete 

circuit of the ecliptic in 19 years. Therefore, in one year it moves 

360° 18 5° 

, or 18.5°. In one month it will move about — : — , or 1.54°. 

19 12 

The moon itself moves 13° per day, as a result of its orbital motion 

13° 
around the earth. Therefore it will move — , or 0.54° per hour. 

24 
1 54 
So the moon will require about -b— hours, or about three hours, 

.54 

to move the distance traveled by the lunar node in a month. 

Hence the difference of three hours between the draconitic and 

sidereal lunar periods. 

Note 38. Stellar Magnitudes (p. 324). 

It is possible to express the light-ratio relations by means of 
very simple formulas. 

Let MijNi be the brightness, or luminosity, of stars of the rath and 
nth magnitudes; and let n be the larger number, belonging to 
the fainter star. 

Then : M _ r^/Joop-m . 

or, passing to logarithms : 

logM. = (i g ^100) (n - ra) = 0.4(ft - ra). 



From this we also obtain 



OKI M 

ft — ra = 2.5 log — 



2e 417 



ASTRONOMY 

These two equations enable us to calculate the light-ratio from 
the difference of magnitudes, and vice versa. 

Note 39. Stellar Photometry (p. 325). 

To understand how this is done, we shall first consider the fol- 
lowing interesting question. What are the faintest stars that can 
be seen with a telescope of given size ? The answer here depends 
on the diameter of the object-glass, because this determines its 
area ; and the area, or light-gathering surface, in turn determines 
the light-gathering power. Now it has been found, by experiment, 
that the faintest star visible in a telescope having an object-glass 
one inch in diameter is of the ninth magnitude. An object-glass 
of diameter d inches will have an area d 2 times as great, and will 
therefore gather d 2 times as much light. Consequently, it will 
just show a star sending us a quantity of light equal to : 

the light of a ninth-magnitude star 
d 2 

If we assume this star to be of the nth. magnitude, we can apply 
the last equation of Note 38. We then have, putting m = 9 : 

M = light of a ninth-magnitude star, 

yy _ light of a ninth-magnitude star 
d 2 
And then our equation gives : 

n- 9 = 2.5 1og^= 2.5 log d 2 ; 

or, n = 9 -f- 2.5 log d 2 . 

This simple equation tells us the magnitude n, of a star just 
visible in a telescope of which the object-glass has a diameter of 
d inches. And it also enables us to calculate the magnitude of a 
star just visible through a diaphragm of which the aperture simi- 
larly has a diameter of d inches. 

Note 40. Light emitted by Vega (p. 326). 

As we have stated, rough measurements show that the entire 
quantity of starlight received by an observer on the earth is equal 
to that of 2000 Vegas. This has also been estimated as being 

418 



APPENDIX 

equivalent to -g-sroVfro-o" of sunlight. Therefore, we receive from 

Vega: 

sunlight y, 1 Qr sunlight , 

33000000 2000 ' 66000000000 ' 

and the word "sunlight" here means the quantity of light received 
from the sun. Then, since the intensity of light diminishes pro- 
portionately to the square of our distance from its source, Vega 
must emit : 

light emitted by sun ^ (distance of Vega) 2 > 
66000000000 (distance of sun) 2 " 

But Vega is one of the stars whose distance has been measured, 
approximately. It has been found that : 

Vega's distance _ ^ §20000 
sun's distance 

Therefore, Vega must emit : 

T . , . ... A , „ 1820000 X 1820000 

Light emitted by sun X r^9 nn „ n , 

8 J 66000000000 ' 

or, approximately : Light emitted by sun X 49. 

Note 41. Motion of Solar System (p. 338). 

Figure 122 may make this matter clearer. The solar system is 
for the moment imagined stationary, and the stars all moving 
with parallel annual velocities represented by the arrows SSi. 





Si. 

122. 


Solar 

o 

System 








s; 


5, 


S 


olar System. 




Fig. 


5 

Motion of S 





On each of these arrows a parallelogram is constructed, having 
one side SSi, directed toward the solar system, or away from it. 
In the two parallelograms shown in the figure, the diagonal velocity 
SSi may be regarded as equivalent to, and it may be replaced by, 
the two smaller velocity arrows forming the sides of the parallelo- 

419 



ASTRONOMY 

grams (cf. p. 399). Only the part SSi affects the velocity of ap- 
proach or recession with respect to the solar system. The entire 
arrow SSi indicates approach on the right-hand side of the figure, 
and recession on the left-hand side. At the lower edge of the 
diagram appears a star none of whose real velocity SSi will appear 
as either approach or recession. 

We might satisfy the above observations if all the arrows SSi 
were replaced by a single parallel arrow, starting from the solar 
system, and pointing toward the right. A study of radial veloci- 
ties all around the sky must therefore prove one of two things : 
either a stream of stars is passing us in a definite direction, or the 
solar system is moving with an equal velocity in the opposite direc- 
tion. The latter hypothesis is, of course, the more probable. 

Note 42. Distance of Vega (p. 341). 

Figure 123 shows the sun, the earth, and Vega. The parallax 

angle, 0".ll, is, 
by definition, the 
angle EVS, sub- 
tended at Vega 
by the radius of 
the earth's orbit 




Fig. 123. Distance of Vega. 



around the sun. As usual, we can solve the right-angled triangle 
ESV, in which we know the angle at .7 and the side ES. We have : 

tan 0^11 = ft, or 57= 



SV[ tan 0".ll 

But tan 0".ll is, approximately : 

0.11 . 
200000 ' 

and so SV, the distance of Vega, is : 

93000000 X 200000 
0.11 

Note 43. Mass of Binary Star (p. 350). 

Referring to Note 261 (p. 405), the formula in the case of a binary 

system is: Mass of system = f^, 

420 



APPENDIX 

where S is the sun's mass, a the linear diameter of the binary- 
orbit in terms of the distance earth-to-sun as a unit, and t the 
binary orbital period in years. 

Note 44. Size of Andromeda Nebula (p. 353). 

Figure 124 will make this clear. S is the sun ; E, the earth ; and 
N, the center of the nebula. C and C are points on the circumfer- 
ence of the nebula. The 
angle ENS is the paral- 
lax of the nebula, here 
assumed to be 0".01 ; 
since it is by definition 
the angle subtended by 
ES, the radius of the 
earth's orbit, to a sup- 
posed observer at the 
nebula. The angle CSC is the angular diameter of the nebula, 
seen from the solar system, and it is 1.5°. Therefore the linear 
distance CC must be greater than ES in the approximate ratio 
1.5 C 




Fig. 124. Size of Andromeda Nebuia. 



0".01 



, or 540,000. 



Note 45. Attraction of Andromeda Nebula (p. 354). 

Regarding the nebula as approximately a globe 540,000 
X 93,000,000 miles in diameter, and the sun as a globe 1,000,000 
miles in diameter, the volume of the nebula equals (540,000 X 93) 3 
times the sun's volume. Let us imagine the sun and nebula to 
have equal densities. Then their masses will be in the above ratio 
of their volumes. But with a parallax of 0".01, the nebula is 
20,000,000 times as far away from us as the sun. Therefore the 
relative attractions of nebula and sun on the earth are : 



(540000 X 93) ! 
(20000000) 2 



or, approximately, 310000000. 



It follows that the nebular density may be as slight as 3i o ooo oir 
of the solar density, and yet the earth be attracted by the neb- 
ula as much as by the sun. 



421 



INDEX 



Aberration of light, 

explanation of, 136 

solar parallax from, 271, 414 
Absolute method of measurement, 331 
Absorption, 

of light by gases, 283 
in the sun, 573 

of starlight by atmosphere, 286 
Adams, discovers Neptune, 248 
Aerolites, 320 

cosmic dust, 320 
Afternoon, not equal to morning, 135 
Airy, Sir Geo., astronomer royal, 31 
Albedo, 218 

of Mercury, 218 

of Venus, 220 
Aldebaran, 

standard first-magnitude star, 324 
Alexandria, Eratosthenes measures earth 

there, 94 
Algol, variable star, 328 
Almanac, 

nautical, 155, 202 

use in finding planets, 50 
Altitude, 36 

American Ephemeris, 202 
Andromeda, constellation, 

nebula in, 353 
size of, 421 
Angle, denned, 29 
Angular diameter, 

of moon, 173 

of planets, 203 

of sun, 118 
Angular distance, defined, 29 
Annular eclipses of sun, 304 
Apex of sun's motion in space, 338 
Aphelion, 263 
Apogee, tides at, 254 
Apollo, ancient name of planet Mercury, 

217 
Apparent orbit, of binary stars, 348 
Apparent solar day, 65 

variable in length, 71 
Apparent solar time, 65 

explanation of, 67 



Apparent solar time, table of differences 

from mean solar time, 82 
Arago, compliments Herschel, 247 
Areas, Kepler's law of, 120, 184, 399 
Aristarchus of Samos, 

his solstice observation used by Hip- 
parchus, 127 
Aristotle, explains lunar phases, 163 
Aristyllus, star observations, 129 
Artificial star, used in photometry, 

325 
Ascension island, Gill's Mars expedition 

to, 266 
Aspect of heavens, on different dates, 72 
Astronomer royal, 

Airy, 31 

Bradley, 136 

office established by Charles II, 159 
Astronomy, 

Chinese, discovery of ecliptic, 29 

definition of the word, 2 

Greek, 30 

popular questions concerning, 2 

value as a study, 21 

value for practical purposes, 18 
Atmosphere, 

absence of, on moon, 166 

extent of terrestrial, from meteor ob- 
servations, 319 

light absorption by, 286 

of earth, 113 

heats meteors, 318 

of Jupiter, 236 

of Mars, 222 

of Mercury, 218 

of Venus, 220 

interferes with transit, 269 

produces refraction, 114 

produces twilight, 113 

retains solar heat, 113 
Auriga, constellation, diagram of, 61 
Aurora, periodicity of, 290 
Average, 

distance between stars, 346 

stellar statistics, 342 
Axial rotation, see Rotation, axial 



423 



INDEX 



Axis, rotation, 

direction in space, planets, 202 

sun, 296 
of celestial sphere, 32 
of earth, 31 

of equatorial mounting, 279 
of telescope mounting, 276 

Balance, torsion, 107 

constant of, 108 
Base-line, for parallax measures, 262 
Bayer, observes Castor and Pollux, 327 
Beard, of comets, 307 
Bessel, measures stellar parallax, 192, 333 
Biela, his comet breaks up, 319 
Binary stars, see Stars, binary 
Blue, color of sky, 113 
Bode's law, 196 

in case of Ceres, 232 
Bond, discovers satellite of Saturn, 246 
Bouguer, measures arc in Peru, 99 
Bradley, J., astronomer royal, 

discovers aberration of light, 136 
Bright-line spectrum, 283 
Bureau of Standards at Washington, 102 

Caesar, his calendar, 138 
Calendar, the, 138 

ecclesiastical, 146, 385 

perpetual, 147 
Campbell, determines apex, 338 

observes Mars, 224 
Canals, of Mars, 223 
Cape of Good Hope observatory, 

heliometer at, 267 
Cassini, computes Horrocks' observation, 

270 
Cassiopeia, constellation, how to find, 54 
Cavendish, weighs the earth, 107, 376 
Cayenne, Richer swings pendulum at, 98 
Celestial equator, precessional motion, 

129 
Celestial meridian, 36 

correspondence with terrestrial, 73, 367 
Celestial poles, 32 

motion of, seen by a traveler, 39 

position with respect to horizon, 40, 
365 
Celestial sphere, 24 

apparent rotation of, 30 

oblique, 42 

parallel, 41 

right, 40 
Center of gravity, 

binary stars, 347 

earth and moon, 174 



Central, 

force, in planetary motion, 187 

sun, 355 
Centrifugal force on the earth, 98 
Century, number in calendar, 143 
Ceres, the first planetoid, 232 
Chaldeans, discover the Saros, 304 
Chamberlin, planetesimal hypothesis, 

358 
Charles II, establishes office of astrono- 
mer royal, 159 
Charting and mapping, 19 
Chemistry, 

of aerolites, 321 

of sun, 286 

and stars, with spectroscope, 284 

stellar, by Huggins, 337 
Chinese astronomy, discovery of ecliptic, 

29 
Chromosphere, of sun, 293 
Chronograph, electric, 278 
Chronometer, 

marine, used in navigation, 157 
earliest, 159 
Church calendar, see Ecclesiastical 

calendar 
Circle, 

diurnal, 33 

ecliptic, 28 

graduated, on sextant, 152 
on telescope, 278, 281 

great, defined, 27 

meridian, 277 
Clerk-Maxwell, 

constitution of Saturn's ring, 245 

light-pressure theory, 309 
Clock, 

astronomic, standard, 278 

of equatorial telescope, 280 

regulator, jeweler's, 279 
Clusters of stars, 351 

distance and size, 352 

nebulous matter in, 352 
Coal-sack in Milky Way, 354 
Collimator, in spectroscope, 282 
Collision, 

of stars, 347 

possible with comet, 308 
Color of sky, 113 
Coma, of comets, 307 
Comets, 14, 307 

designation of, 311 

capture theory of, 314 

light-pressure theory of tails, 309 

periodic, 313 
Compound lenses of telescope, 273 



424 



INDEX 



Conic sections, Newton's comet orbits, 

312 
Conjunction, 209 

superior and inferior, 210 

of Venus, produces transit, 268 
Conservation of energy, 2 

effect on tidal friction, 256 
Constant, 

of torsion balance, 108 

in calendar calculations, 144 
Constellations, 7 

diagrams of principal, 63 
Continuous spectrum, 283 
Cooling of stars, 6 
Copernicus, 

his book De Revolutionibus, 87 

his planetary theory, 191 
Cordova catalogue of stars, 334 
Corona, solar, 295 

Correction, of sextant observations, 156 
Cosmic dust, 320 
Cosmic velocity, 

of solar system, 338 

of stars, 346 
Cosmogony, 356 
Council of Nice^46 
Crests, of tidal waves, 253 
Cross-threads, in telescope, 275 
Crystal sphere, in Ptolemaic theory, 189 
Curvature, of earth, 

arguments proving, 87 

measurement of, 97 
Curves in planetary motion, 215 
Cygnus, constellation, diagram of, 61 

Darwin, theory of moon's origin, 258 
Date, in calendar, 

four parts of, 138 

of Easter, 148 
Date-line, international, 75 
Day, 65 

apparent solar, 67 

lengthening of, by tidal frictions, 257 

lunar, 176 

midsummer and midwinter, 121 

on Mars, 221 
Mercury, 218 
Venus, 221 

planetary, 202 

sidereal, 66 

solar, unequal, 71 
Day and night, 31 

at the pole, 42 

equal at equator, 40 

in temperate regions, 43 

longest and shortest, 121 



Dead-reckoning, in navigation, 151 
Declination, defined, 34, 363 

measured, 278 
Deferent, in Ptolemaic theory, 189 
Degree of latitude, varying length, 97 
Deimos, satellite of Mars, 222 
Demon star, see Algol 
Density, 

comets, 308 

earth, 110 

moon, 175 

nebulae, 354 

stellar distribution, 345 

sun, 292 
Departure, in navigation, 151 
Diameter, 

angular, of moon, 172 
sun, 118, 291 
planets, 203 

of planets, in miles, 204 

of sun, in miles, 291 
Differences of time, 72 

sidereal and solar, 73 
Differential method of measurement, 332 
Dipper, constellation, see Ursa Major 
Disks, planetary, 13 

seen in field glass, 52 
Distance, see also Parallax 

change of stellar, observed with spec- 
troscope, 284 

of star clusters, 352 

of stars, 322, 330 

of Vega, 420 
Diurnal, 

circles, 33 

inequality of tides, 253 

observations of Mars for solar parallax, 
263 
Doppler principle, with spectroscope, 284 
Double 

stars, 8. See also Stars, binary 

telescopes, photographic, 281 

double star, 351 
Douglass, observations of Mars, 227 
Draconitic period, 305, 417 

Earth, 

an astronomic body, 15 
atmosphere of, 113 
curvature of, 87, 97 
density of, 110 
flattening of, 97 
interior of, 111 
measurement of, 92, 95 
orbit, form of, 116 
rotation, 15, 30, 88 



425 



INDEX 



Earth, shadow of, in eclipses, 303 

shape a geoid, 101 

weighing it, 103 
Earth-shine on moon, 164 
Easter, date of, 148, 385 
Eccentricity of planetary orbits, 200 
Ecclesiastical calendar, 146 
Eclipses, 297 

annular, of sun, 304 

lunar, 301 

solar, 297 

periodicity and Saros, 304 

umbra and penumbra, 303 

variable star, 328 
Ecliptic, 26 

locating it on sky, 47 

pole of, 131 
Egyptians, find length of year, 127 
Electric chronograph, 278 
Elements, 

chemical, in sun, 284, 286 
in aerolites, 321 

of orbits, binary stars, 348 
comets, 313 
planets, 201 

perturbations, 206 
Elongation, 

maximum, of planets, 212, 410 

of satellites from planets, 205 
Energy, conservation of, 2 

meteors, 318 

tidal friction, 256 
Ephemeris, planetary, 202 
Epicycle, in Ptolemaic theory, 189 
Equation of time, 134 

table of, 82 
Equation, personal, see Personal error 
Equator, 

terrestrial and celestial, 33 

precessional motion of, 129 
Equatorial, mounting of telescopes, 279 
Equinoxes, 35, 43, 364 

precession of, 126 
Eratosthenes, 

measures ecliptic, 30 
size of earth, 92 

observes j3 Librae, 327 
Erecting lens, in telescope, 273 
Eros, planetoid, its orbit, 236 

observed for solar parallax, 267 
Evolution, tidal, 256 
Eye-piece, in telescope, 273 

Faculae, of sun, 289 

Fixed stars, see Stars 

Flagstaff, Lowell observatory at, 227 



Flash spectrum, in solar eclipse, 288 
Flattening, 

of earth, 97 

of planets, 205 

of planets' orbits, 200 
Focus, 

of earth's orbit, 116 

of telescope, 272 
Force, 

centrifugal, on earth, 98 

central, in planetary motion, 187 

tidal, 252 

repulsive, of sunlight, 309 
Foucault experiment, 89, 371 
Fraunhofer, spectrum lines, 287, 308 
Friction, tidal, 256 

Galaxy, see Milky Way 
Galileo, 

advocates rotation of earth, 88 

discovers Jupiter's satellites, 237 

his telescope, 273 

observes Saturn's ring, 241 
sunspots, 17 

phases of Venus, 219 
Galle, discovers Neptune, 248 
Gauss, 

computes orbit of Ceres, 233 

Easter date, 148, 385 
Gemini, constellation, diagram of, 61 
Geodesy, 95 
Geography, 

latitude and longitude in, 34 

terrestrial meridians in, 73 
Gtoid, shape of earth, 101 
Georgium Sidus, 

name for Uranus, 247 
Gill, 

observes Mars for solar parallax, 266 
Globe, celestial, 37 

use of, 63 
Gnomon, of sundial, 78 

construction of, 368 
GSttingen, observatory, heliometer, 267 
Graduated circle, 

of sextant, 152 

of telescope, 281 
Gravitation, 

action of, inside nebulae, 4 

force of, on sun, 102, 291 

Newton's law of, 103, 184 

proves distance of stars, 322 
Great Bear, constellation, see Ursa 

Major 
Great circle of sphere, defined, 27 
Greatest luminosity of Venus, 219 
426 



INDEX 



Greenwich, initial meridian, 34, 73 
Gregorian calendar, 138 
Groombridge, 

catalogue of stars, 334 

his runaway star, 347 

Hale, spectroheliograph, 294 
Halley, 

his comet, 311 

transit of Venus method, 269, 412 
Harmonic law of Kepler, 188, 400 
Harvest moon, 177 
Heat, 

interior of earth, 111 

of meteors, cause of, 318 

of stars, 326 

solar, retained by atmosphere, 113 
Heavens, aspect on different dates, 72 
Height of meteors, 319 
Heliacal rising of stars, 127 
Heliometer, 267 

Helmholtz, theory of solar energy, 292 
Hemisphere, southern, seasons, 122 
Herschel, Captain John, perpetual calen- 
dar, 147 
Herschel, Sir John, 

distances in solar systems, 12 

star magnitudes, 323 

V Argus observation, 327 
Herschel, Sir William, 

apex of sun's way, 339 

discovers satellite of Saturn, 246 

discovers satellite of Uranus, 247 

discovers Uranus, 232 

explains Galaxy, 354 

star counts, 355 
Hipparchus, 

explains eclipses, 297 

observes solstice, 127 

originates Ptolemaic theory, 189 

scale of star magnitudes, 323 
Honolulu expedition, 112 
Hooke, corresponds with Newton, 91 

ideas about comets, 312 
Horizon, defined, 36 
Horns, of moon, 165 
Horrocks, 

observes transit of Venus, 270 
Hottest day of summer, 122 
Hour-angle, 

defined, 66, 363 

measures time, 66 

relation to sidereal time, 366 
Hour-circle, defined, 363 
How to know the stars, 45 
Huggins, spectroscopist, 336 



Hutton, calculates Maskelyne's observa- 
tions, 104 
Huygens, 

Saturn's ring, 13, 241 
Hydrocarbon, in comets, 308 
Hydrogen jets from sun, 293 

Ice age, geologic, 125 

Image, focal, in telescope, 272 

Imperfections of visual observations, 228 

Inclination, 

of lunar orbit, 160, 298 

of planetary orbit, 200 
Inequality, 

diurnal, of tides, 253 

of morning and afternoon, 135 
Inferior planets, 209 

conjunctions, 210 

oscillations of, 213 

period of, 409 
Inhabitants of Mars, 223 
Instruments, astronomic, 272 
International date-line, 75 
Invariable plane, 207 
Iris, observed by Gill, 267 

Janssen, observes prominences, 293 
Julian calendar, 138 
Juno, planetoid, 234 
Jupiter, 

appearance in telescope, 13 

atmosphere, 236 

comet family of, 314 

distance from sun, 240 

how to find, 51 

influence on planetoid orbits, 235 

in nebular hypothesis, 358 

longitude from satellite observations, 
239 

markings on, 236 

rotation, axial, 236 

satellites, 237 

seasons and temperature, 237 

Kapteyn, stellar researches, 342 

Keeler, 

constitution of Saturn's ring, 245 
planetesimal hypothesis, 359 
spiral nebulae, 353 

Kepler, 

ellipticity of earth's orbit, 116 
harmonic law, 188, 400 
ideas about comets, 312 
law of areas, 120, 184, 399 
laws apply to binary stars, 348 
variation in distance from earth to 
sun, 193 



427 



INDEX 



Kinetic theory of gases, 

explains absence, of lunar atmosphere, 
167 
of Martian atmosphere, 222 
in nebular hypothesis, 357 
similar to cosmic stellar theory, 347 
Kiistner, latitude variation, 112 

La Condamine, Peru arc, 99 
Lagrange, planetoid orbits, 235 
Land-fall, in navigation, 151 
Laplace, 

capture theory of comets, 314 

nebular hypothesis, 235, 356 
binary stars, 350 
satellites of Uranus, 247 
Lapland, arc measured in, 99 
Lassell, satellites of Uranus, 247 
Latitude, 

arcs of, used in r. odesy, 99 

found in navigation, 154 

terrestrial, 34 

variation of, 112 
Lava, source of, 112 
Law, Bode's, 196 

Kepler's, 187, 399 

Newton's, 103, 184 
Layer, reversing, in sun, 288 
Leap-year, rule for, 142 
Leipzig observatory, heliometer, 267 
Lens, telescopic, compound and erecting, 

273 
Leo, constellation, diagram of, 62 
Leonid meteors, 316 
Leverrier, discovers Neptune, 248 
Lexell, explains Uranus, 247 
Libration, 

of moon, 171 

of Mercury, 218 
Light, 

aberration of, 136 

solar parallax, 271, 414 

absorption of, by gases, 283 
terrestrial atmosphere, 325 

gathering power of telescopes, 275 

of meteors, cause of, 318 

of sun, source of, 292 

total, of stars, 325 

velocity of, 333 
Light-pressure in comet tails, 309 
Light-ratio of star magnitudes, 324 
Light-year and stellar parallax, 333 
Limits, in eclipses, 300 
Line-of -sight, 

motions of stars in, 334 

of telescope, 277 



Lockyer, observes prominences, 293 
Logogriph of Huygens, 241 
Long Island Sound, tides in, 256 
Longitude, 

arcs of, used in geodesy, 99 
determined from Jupiter's satellites, 
239 
occultations, 239 
in navigation, 157 
terrestrial, 34, 73 
Lowell,, 

markings on Mercury, 218 

Venus, 221 
Mars observations, 226 

Magnetic storms, periodicity of, 290 
Magnifying power of telescopes, 274 
Magnitudes of stars, 323 
Mapping and charting, 19 
Maps of stars, 45 
Markings, 

on Mercury, 218 
Mars, 223 
Jupiter, 236 
Venus, 221 
Mars, 

how to find, 51 
inhabitants, 223 
observed for solar parallax, 263 
Maskelyne, weighs the earth, 104 
Mass, 

distinction from weight, 102 
of Algol, 329 

binary stars, 350, 420 
comets, 308 
earth, 110 
moon, 173, 396 
planets, 204, 405 
sun, 291, 415 
Matter, 2 

Maupertuis, arc in Lapland, 99 
Maxwell, J. Clerk, 
light-pressure, 309 
Saturn's ring, 245 
Mean solar day, 65, 71 
Mean solar time, 71 

table of difference from apparent time, 
82 
Mercury, 217 
how to find, 50 
transits of, 306 
Meridian, 

celestial, 36, 73 

circle, 276 

distance of stars from, 67 

of Greenwich, 34 



428 



INDEX 



Meridian, 

planets on it at midnight, 52 

right-ascension of, 67, 366 

shape of terrestrial, 97 

standard, 74 
Meteors, 315 
Micrometer, 276 

used for binary stars, 347 
Mars, 265 
stellar parallax, 332 
Midsummer and midwinter day, 121 
Milky Way, 354 
Minor planets, see Planetoids 
Mira, variable star, 328 
Month, lunar, in eclipses, 298 
Moon, 

absence of atmosphere, 17, 166 

always near ecliptic, 49, 160 

angular velocity of apparent motion, 
161 

axial rotation, 169 

causes tides, 252 

craters and mountains, 17, 181 

Darwin's theory of its origin, 258 

density, 175 

dimensions, 16 

distance, 16, 169, 394 

draconitic period, 305, 417 

earth-shine on, 164 

eclipses, 164, 297 
limit, 300 
where visible, 303 

effect of tides on, 257 

harvest, 177 

horns turn away from sun, 165 

its attraction produces precession, 130 

libration, 171 

lunar day, 176 

measurement of mountains on, 182 

measurement of size of, 173 

month of, 298 

not self-luminous, 16, 160 

occults stars, 166 

orbit plane, inclination of, 160 

orbit revolution, 16, 160 

our nearest neighbor, 16, 160 

parallax of, 169, 396 

perigee of, 169 

phases, 16, 162 
when eclipsed, 301 

physical appearance, 16 

rides high, 179 

rising, variation in time of, 176 

shape of orbit, 168, 181, 397 

sidereal and synodic periods, 161 

surface features, 16 



volume, 173 

weight or mass, 173, 396 
Morning and afternoon unequal, 135 
Morning and evening stars, 22 
Moulton, planetesimal hypothesis, 358 
Mountain, 

used to weigh the earth, 104 

measurement of lunar, 182 
Mountings, for telescopes, 276 
Multiple stars, 351 

Nautical almanac, 155, 202 
Navigation, 

an astronomic process, 20 

dead-reckoning in, 151 

"departure" in, 151 

early method of, 159 

finding ship's latitude, 154 

fundamental problem of, 151 

nautical almanac in, 155 

sextant in, 152 
Neap tides, 255 
Nebulae, 

density, 354 

effect of internal gravitation, 4 

gaseous constitution of, 4 

in Andromeda, 353, 421 

in star clusters, 352 

Laplace's hypothesis, 235, 356 

nebulium in, 353 

number of, 5 

planetary, 352 

resolving them, 3 

ring form, 353 

spiral form predominant, 5, 353 

spiral, in planetesimal hypothesis, 359 
Nebular hypothesis, 235, 356 

satellites of Uranus in, 247 
"Nebulium" in nebulae, 353 
Nebulosity of cometary coma, 307 
Neptune, planet, 247 
New Haven observatory, 267 
New stars, see Temporary stars 
Newton, 

comet orbits, 312 

determines flattening of earth, 98 

law of gravitation, 103, 184 

test of earth's rotation, 91 
Nice, council of, 146 
Night, radiation from earth in, 122 
Nodes, 

motion of lunar, in eclipses, 299 

of Milky Way, 354 

planetary orbital, 200 

transit of Venus in, 268 
Nodules, of sun, 289 



429 



INDEX 



Novce, see Temporary stars 
Nucleus, 

in development of binary stars, 350 

of comets, 307 
Nutation, of terrestrial axis, 132 

Object-glass, of telescope, 272 
Oblique sphere, 42 
Observations, 

correction of, 156 

imperfections of visual, 228 

planetary, for orbit determination, 199 
Occultations, of stars, 166 

used for determining longitude, 239 
Olbers, telescopic constellations, 310 
Opposition, of planets, 212 

of Mars, favorable, 263 
Orbit, binary star's, 347 

cometary, 312 

earth's, around sun, 25, 116 
slow changes of, 125 

eccentricity of, 200 

elements of, six, 201 

inclination of, 200 

meteoric, 316 

moon's, shape of, 168, 181, 397 

nodes, line of, 200 

perturbations of, 206 

planetary, 197 

in nebular hypothesis, 356 

in planetesimal hypothesis, 358 

stability of, 206 

of planet's satellites, 205 

stellar parallactic, 331 
Orientation of sundial, 82 
Orion, constellation, diagram of, 62 
Oscillations, inferior planets, 213 

tidal, 253 

Pallas, planetoid, 233 

Pantheon, Foucault experiment, 89 

Parallax, lunar, 169, 396 

solar, aberration of light, 271, 414 

definition of, 260 

definitive value of, 268 

diurnal method, 263 

Gill's observations, 266 

Mars observations, 263 

perturbation method, 271 

scale of solar system, 262 

transit of Venus, 268 
stellar, averages, Kapteyn, 345 

Bessel's, 192 

clusters, 352 

defined, 192, 330 

light-year, 333 



measurement of, 331 
orbits, 331 

photographic method, 332 
relative, 332 
Parallel sphere, 41 
Pendulum, Foucault's experiment, 89 

Bicher's, at Cayenne, 98 

shape of earth from, 101 
Penumbra, eclipse shadow, 303 
Perigee, lunar, 169 

tides, 254 
Perihelion, defined, 120 

earth in, 123 

passage, time of, 201 
Period, direct observation of planetary, 
192, 215 

Draconitic, 305, 417 

Saros in eclipses, 304 

planetary, in Kepler's laws, 188 
orbital element, 201 

sidereal and synodic, lunar, 161 
planetary, 208, 408 

table of approximate planetary, 211 
Periodic changes in earth's orbit, 125 
Periodic comets, 313 
Periodic perturbations, 206 
Periodic variable stars, 328 
Periodicity, magnetic storms, etc., 290 

meteor showers, 316 

recurrence of eclipses, 304 

sunspots, 290 
Perpetual calendar, 147 
Perseid meteors, 316 
Personal error, 266 
Perturbations, by stars, 322 

by comets, 308 

of planetary orbits, 206 

of planetoid orbits, 235 

solar parallax from, 271 
Peru, terrestrial arc measured in, 99 
Phase, 11 

lunar, 162 

effect on tides, 255 
in eclipses, 301 

Mars, 222 

Mercury, 218 

Saturn's ring, 242 

Venus, 218 
Phobos, inner satellite of Mars, 222 

in nebular hypothesis, 357 
Photographic observations, Mars, 229 

planetoids, 234 

stellar magnitudes, 325 

stellar parallax, 332 

stellar spectra, 336 
Photographic telescopes, 281 



430 



INDEX 



Photometer, stellar, 324, 418 
Photosphere, solar, 288 
Piazzi, discovers Ceres, 232 
Pickering, spectroscopic binary stars, 

349 
Plane, invariable, 207 
Planetesimal hypothesis, 358 
Planetoids, 183, 196, 231 

Ceres, the first one, 232 

Eros, 236 

mass and size, 235 

Pallas, Juno, Vesta, 233 

Wolf, photographic discovery, 234 
Planets, axial rotation period, 202 

brilliancy, 46 

curves in motion of, 215 

disks visible in telescope, 13 

elongation from sun, 211 

field-glass view of, 52 

identification of, 47 

inferior, 209 

mass measured, 204, 405 

morning and evening stars, 22 

motion among stars, 10 

names, 10, 183 

near ecliptic always, 47 

not self-luminous, 11 

on meridian at midnight, 52 

opposition of, 212 

orbital elements, 200 

orbits determined, 197 

oscillations of, 213 

periods in Kepler's laws, 188 

phases, 11 

planetesimal hypothesis, 359 

proximity to us, 10 

revolution around sun, 10, 183 

rotation poles of, 203 

retrograde motions, 214 

sidereal period, 207, 408 

size measured, 203 

superior, 209 

surface and volume measured, 204 

synodic period, 207, 408 

twinkling, 50 

ultra-neptunian, 249 

visibility of, 211 
Planisphere, 63 
Pleiades, motion in, 336 
"Pointers," constellation, see Ursa Major 
Polar axis, in telescope mounting, 280 
Poles, celestial and terrestrial, 32 

motion of, seen by travelers, 39 

rotation of celestial, 132 

of ecliptic, 131 

planetary, position of rotation, 203 



position of, above horizon, 40, 365 

precessional motion of celestial, 131 
Pole star, effect of precession on, 132 

how to find, 53 
Position angle, binary stars, 348 
Power of telescope, light-gathering, 275 

magnifying, 273 
Precession of equinoxes, 126 

cause of, 129 

changes right-ascension, etc., 334 

determines date of pyramids, 133 

effect on pole star, 132 
Prime meridian, Greenwich, 34, 73 
Prism, in spectroscope, 282 
Proctor, motion of "Dipper" stars, 335 
Prominences, solar, 293 
Proper motion, of stars, 334 

determines apex, 339 

in Pleiades, 351 
Ptolemy, phases of Venus, 219 

planetary theory, 189 
Pyramid, date of construction, 133 
Pythagoras, earth's motions, 87 

Radial velocity, stellar, 334 

determines apex, 338 
Radiant, of meteor showers, 315 
Radiation of heat from earth, 122 
Radius vector, 119 

law of areas, 120, 184 
Rate of chronometers, 157 
Recurrence, of eclipses, 304 

of meteor showers, 316 
Refraction, atmospheric, 114 

correction of sextant observations, 156 
"Regulator" clocks, 279 
Relative stellar parallax, 332 
Retrograde motions of planets, 214 
Reversing layer, in solar spectrum, 288 
Richer, Cayenne observations, 98 
Right-ascension, 34, 363 

measured, 278 

of meridian, 67, 366 

sidereal time and hour-angle, 367 
Right sphere, 40 
Rigidity of earth, 112 
Ring nebulae, 353 
Ring of Saturn, 13, 241, 412 

constitution of, 245 

disappearance of, 244 

Keeler and Maxwell, 245 

phases, 242 
Rising and setting, 31 

heliacal, 127 

of moon, 176 
Roemer, observes Jupiter's satellites, 239 



431 



INDEX 



Rotation, axial, celestial sphere, 30 

earth, 15, 3Q 

Foucault experiment, 89 

Jupiter, 236 

Mars, 221 

Mercury, 218 

moon, 169 

Newton's experiment, 91 

planets, 202 

position of poles, 203 

sun, 295 

tidal effect on earth's, 253 

Venus, 221 
Runaway star, 335 

Sagredus, character in Galileo's Dialogue, 

88 
Salusbury, translator of Galileo, 88, 237 
Salviati, character in Galileo's Dialogue, 

88 
Sappho, planetoid, 267 
Saros, eclipse period, 304 
Satellites, 

distance from planets, 205 

eclipses, 238 

in planetesimal hypothesis, 360 

Jupiter, 237 

longitude from observing them, 239 

Mars, 222 

Saturn, 246 
Saturn, appearance in telescope, 241 

how to find, 51 

moons, 246 

ring, 13, 241, 412 
Scale, of solar system, 262, 381 

of stellar system, 346 
Schehallien, mountain in Scotland, 104 
Schiaparelli, meteor and comet orbits, 

319 
Schwabe, periodicity of sunspots, 290 
Scintillation, see Twinkling 
Scorpius, constellation, diagram, 62 
Seasons, explanation, 44, 120 

Mars, 222 

Mercury, 217 

Jupiter, 237 
Secchi, stellar chemistry, 337 
Secular perturbations, 206 
Seeing, process of, 228 
Semi-diameter, in sextant observing, 156 
Semi-diurnal tides, 253 
Sextant, in navigation, 152 

theory of, 393 
Shadow, of earth in eclipses, 303 
Shooting stars, see Meteors 
Showers of meteors, 315 



Sidereal, 

day, 65 

period, moon, 161 
planets, 207, 408 

space, unit of, 345 

time, 65, 366 

year, 128 
Sight line, of telescope, 277 

stars' motion in, 334 
Sirius, magnitude, 324 

member of Dipper group, 336 

velocity observed by Huggins, 336 
Sky, color, 113 

definition, 22 
Slough, Herschel at, 247 
Solar system, chance of reaching Vega, 
341 

cosmic motion, 338, 419 

future of, 361 

in planetesimal hypothesis, 360 
Solar time, see Apparent solar time 
Solstice, observed to get length of year, 
126 

summer, 93, 121 

winter, 121 
Sosigenes, arranges Julian calendar, 138 
Southern hemisphere, conditions there, 

43, 122 
Space, 2 

unit of sidereal, 345 
Specific gravity of earth, 110 
Spectroheliograph, 294 
Spectroscope, 282 

chemistry of sun and stars, 284 

Doppler principle, 284 

radial velocities, 335 

slitless, 285 

solar prominences, 293 

used for binary stars, 349 
Mars, 224 
Nebulae, 4 
Saturn's ring, 245 
Spectrum, 

bright-line and continuous, 283 

classification of stellar, 337 

cometary, 308 

flash, in solar eclipses, 288 

Fraunhofer lines in solar, 287 

reversing layer in solar, 288 

shift of lines in, 284 
Sphere, celestial, 23 

apparent rotation, 30 

oblique, 42 

parallel, 41 

right, 40 

crystal, in Ptolemaic theory, 189 



432 



INDEX 



Spiral nebulae, 5 

Spots on the sun, 17, 289 

Spring tides, 255 

Stability, of planetary orbits, 206 

Stadium, Greek linear measure, 94 

Standard clocks, astronomic, 278 

Standard magnitudes, stellar, 324 

Standard meridians, 74 

Standard pound, 102 

Standard time, 65, 74, 83 

Stars, analogy to sun, 6, 322 

artificial, for photometry, 325 

average distance asunder, 346 

binary, 347 

change in distance of, spectroscopic, 284 

chemistry of, spectroscopic, 284 

clusters, 351 

collision, 347 

community of motions, 335 

cooling, 6 

cosmic velocity, 346 

dates when on meridian at 9 p.m., 58 
rising and setting, 9 p.m., 60 

density of, in sidereal space, 345 

distance, 330 

double, 8 

excessively remote, 322 

faintest visible in telescope, 418 

fixed, 7, 333 

heat of, 326 

Herschel's comets, 355 

Huggins, their chemistry, 337 

identifying them, 52 

Kapteyn's statistical researches, 342 

kinetic theory of, 347 

light-ratio of, 324 

line-of-sight motions, 334 

list of brightest, 57 

magnitudes, 6, 323 

maps, 45 

Milky Way, 354 

morning and evening, 22 

novoe, or new stars, 8, 327 

number visible to eye, 7 

originate in nebulae, 4 

parallax, 330 

periodically variable, 328 

points of light only, 12 

proper motions, 334 

radial velocity, 334 

runaway, 335 

self-luminous, 6 

spectra photographed, 336 

streams of, 347 

subject to gravitation, 7 

total light of, 325 



twinkling, 6 

variation of brightness, 8, 326 
in clusters, 351 
Stationary points, in planetary motion, 

215 
Statistics of stars, 342 
Storms, magnetic, periodicity, 290 

solar, 289 
Streams, stellar, 347 
Summer, heat of, 120 

longer than winter, 123 
Sun, absorption in outer layers of, 286 

analogy to stars, 6, 322 

angular diameter, 118 

annular eclipses, 304 

apex of its cosmic motion, 338 

axial rotation, 17, 295 

central, 355 

chemistry of, 286 

chromosphere, 293 

corona, 295 

density, 292 

dimensions, 17, 290 

direction of rotation axis, 296 

distance, from Jupiter's satellites, 240 
scale of solar system, 262 
see Parallax 

eclipses, 297 

eclipse limits, 300 

effect on tides, 254 

faculae, 289 

focus of earth's orbit, 116 

Fraunhofer lines, 287 

gravity on, 102, 291 

heats earth, 121 

mass, 291, 415 

motion, in ecliptic, 29 
in space, 7, 338 

nodules, 289 

photosphere, 288 

planetesimal hypothesis, 360 

position on sky, 25, 27 

prominences, 293 

reversing layer, 288 

semi-diameter correction, 156 

source of light and heat, 292 

spots, 17, 289 

stellar magnitude, 324, 326 

twin suns, 347 

volume, 292 
Sundial, 78 

mathematics of, 368 
Sunspots, 17, 289 

possible stellar, 328 
Superior, conjunctions, 210 

planets, 209 



2f 



433 



INDEX 



Surface, area of planets, 204 

of aerolites, 321 
Syene, Eratosthenes observes at, 94 
Synodic period, moon, 161 

planets, 208, 408 

Tail, comets, 308 

meteors, 315 
Telescope, 272 

cross-threads, 275 

eye-piece, 273 

equatorial, 279 

magnifying power, 274 

mounting, 276 

object-glass, 272 

photographic, 281 

sweeping sky with, 310 
Temperature, depends on day and night, 
120 

increase inside earth, 111 

Jovian, 237 

Martian, 226, 411 
Temple's comet, 319 
Temporary stars, 8, 326 
Terrestrial telescopes, 273 
Tides, apogee, and perigee, 254 

caused by moon, 251 

diurnal inequality, 253 

effect of sun on, 254 

effect on binary stars, 350 

effect on moon, 257 

evidence of solidity of earth, 111 

inequality of, 253 

Long Island Sound, 256 

oscillations, 253 

planetesimal hypothesis, 358 

semi-diurnal, 253 

situation of crests, 253 

spring and neap, 255 

tidal evolution and friction, 256 
Time, 

apparent solar, 67 

determined by observation, 279 

differences, 72 

equation of, 134 

mean solar, 71 

sidereal, 65 

standard, 65, 74 

sundial, 82 
Timocharis, 129 
Torsion balance, 107 

constant of, 108, 375 
Transit, of Mercury, 306 

of Venus, 221, 268, 306, 412 
Triangulation, geodetic, 95 
Tropical year, 128 



length of, 141 

used in calendar, 141 
Tuttle, comet, 319 
Twilight, 113 
Twinkling, planets, 50 

stars, 6 
Tycho Brahe, Kepler uses his observa- 
tions, 194 

observes comet of 1577, 311 

observes temporary star, 327 

Ultra-Neptunian planets, 349 
Umbra, eclipse shadow, 303 

sunspots, 290 
Unit, mass and weight, 102 

modern system, 108 

natural and artificial, 140 

sidereal space, 345 

time, 65 
Universe, 1, 356 
Uranus, 246 

Ursa Major, constellation, community of 
motion in, 336 

how to find, 52 

Variable stars, 8, 326 

in clusters, 351 

periodic, 328 
Variation of latitude, 112 
Vega, distance of, 420 

light emitted by, 418 

near apex, 338 

possibility of solar system reaching it, 
341 
Velocity, cosmic, of solar system, 338 

of stars, 346 

of light, 333 

of " runaway " star, 347 
Venus, 14, 218 

atmosphere, 220 

attains greatest luminosity, 219 

how to find, 51 

markings on, 221 

phases, 218 

transit, 221, 268, 306, 412 
Vernal equinox, 35, 72 
Vertex of angle, defined, 29 
Vesta, planetoid, 234 
Victoria, planetoid, 267 
Visibility, comets, 311 

of objects by sunlight, 113 

planets, 211 
Vogel, eclipse theory of Algol, 328 

photographs stellar spectra, 336 
Volcanoes, 112 



434 



INDEX 



Volume, comets, 307 
moon, 173 
planets, 204 
sun, 292 

Wave motion, tidal, 255 
Week, calculation of day of, 143 
Weight, distinction from mass, 102 

of earth, 103 

of moon, 173 
Winter, cold of, 120 

shorter than summer 123 

southern hemisphere, 124 



Witt, discovers Eros, 236 

Wolf, photographs planetoids, 234 

Year, ancient methods of determining its 
length, 126 
in chronology, 140 
number of days in it, 70 
sidereal and tropical, 128 
synodic, 208 

Zenith, 36 

stars brightest near, 325 
Zero, a stellar magnitude, 324 
Zodiacal light, 249 



435 



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